id sid tid token lemma pos cana-3316 1 1 communications communication NOUN cana-3316 1 2 on on ADP cana-3316 1 3 applied apply VERB cana-3316 1 4 nonlinear nonlinear ADJ cana-3316 1 5 analysis analysis NOUN cana-3316 1 6 issn issn NOUN cana-3316 1 7 : : PUNCT cana-3316 1 8 1074 1074 NUM cana-3316 1 9 - - PUNCT cana-3316 1 10 133x 133x NUM cana-3316 1 11 vol vol NOUN cana-3316 1 12 32 32 NUM cana-3316 1 13 no no NOUN cana-3316 1 14 . . PUNCT cana-3316 2 1 6s 6s NUM cana-3316 2 2 ( ( PUNCT cana-3316 2 3 2025 2025 NUM cana-3316 2 4 ) ) PUNCT cana-3316 2 5 529 529 NUM cana-3316 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3316 2 7 power power NOUN cana-3316 2 8 dominator dominator NOUN cana-3316 2 9 equitable equitable ADJ cana-3316 2 10 coloring color VERB cana-3316 2 11 for for ADP cana-3316 2 12 some some DET cana-3316 2 13 standard standard ADJ cana-3316 2 14 graphs graph NOUN cana-3316 2 15 g. g. PROPN cana-3316 2 16 navamani navamani PROPN cana-3316 2 17 1 1 NUM cana-3316 2 18 , , PUNCT cana-3316 2 19 l. l. PROPN cana-3316 2 20 jacquline jacquline PROPN cana-3316 3 1 2 2 NUM cana-3316 3 2 * * SYM cana-3316 3 3 1,2 1,2 NUM cana-3316 3 4 * * NOUN cana-3316 3 5 saveetha saveetha PROPN cana-3316 3 6 school school NOUN cana-3316 3 7 of of ADP cana-3316 3 8 engineering engineering PROPN cana-3316 3 9 , , PUNCT cana-3316 3 10 saveetha saveetha PROPN cana-3316 3 11 institute institute PROPN cana-3316 3 12 of of ADP cana-3316 3 13 medical medical ADJ cana-3316 3 14 and and CCONJ cana-3316 3 15 technical technical ADJ cana-3316 3 16 sciences sciences PROPN cana-3316 3 17 saveetha saveetha PROPN cana-3316 3 18 university university PROPN cana-3316 3 19 , , PUNCT cana-3316 3 20 chennai chennai NOUN cana-3316 3 21 602105 602105 NUM cana-3316 3 22 , , PUNCT cana-3316 3 23 tamil tamil PROPN cana-3316 3 24 nadu nadu PROPN cana-3316 3 25 , , PUNCT cana-3316 3 26 india india PROPN cana-3316 3 27 . . PROPN cana-3316 3 28 1 1 NUM cana-3316 3 29 email email NOUN cana-3316 3 30 : : PUNCT cana-3316 3 31 g_navamani@yahoo.co.in g_navamani@yahoo.co.in PROPN cana-3316 3 32 , , PUNCT cana-3316 3 33 2 2 NUM cana-3316 3 34 * * NOUN cana-3316 3 35 email email NOUN cana-3316 3 36 : : PUNCT cana-3316 3 37 jackinfanci@gmail.com jackinfanci@gmail.com PROPN cana-3316 3 38 article article PROPN cana-3316 3 39 history history NOUN cana-3316 3 40 : : PUNCT cana-3316 3 41 received receive VERB cana-3316 3 42 : : PUNCT cana-3316 3 43 23 23 NUM cana-3316 3 44 - - SYM cana-3316 3 45 10 10 NUM cana-3316 3 46 - - PUNCT cana-3316 3 47 2024 2024 NUM cana-3316 3 48 revised revise VERB cana-3316 3 49 : : PUNCT cana-3316 3 50 07 07 NUM cana-3316 3 51 - - SYM cana-3316 3 52 12 12 NUM cana-3316 3 53 - - PUNCT cana-3316 3 54 2024 2024 NUM cana-3316 3 55 accepted accept VERB cana-3316 3 56 : : PUNCT cana-3316 3 57 16 16 NUM cana-3316 3 58 - - SYM cana-3316 3 59 12 12 NUM cana-3316 3 60 - - PUNCT cana-3316 3 61 2024 2024 NUM cana-3316 3 62 abstract abstract NOUN cana-3316 3 63 : : PUNCT cana-3316 3 64 a a DET cana-3316 3 65 power power NOUN cana-3316 3 66 dominator dominator NOUN cana-3316 3 67 coloring coloring NOUN cana-3316 3 68 of of ADP cana-3316 3 69 a a DET cana-3316 3 70 graph graph NOUN cana-3316 3 71 𝐺 𝐺 NOUN cana-3316 3 72 is be AUX cana-3316 3 73 a a DET cana-3316 3 74 proper proper ADJ cana-3316 3 75 coloring coloring NOUN cana-3316 3 76 where where SCONJ cana-3316 3 77 each each DET cana-3316 3 78 vertex vertex NOUN cana-3316 3 79 in in ADP cana-3316 3 80 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 3 81 ) ) PUNCT cana-3316 3 82 power power NOUN cana-3316 3 83 dominates dominate VERB cana-3316 3 84 at at ADP cana-3316 3 85 least least ADV cana-3316 3 86 one one NUM cana-3316 3 87 complete complete ADJ cana-3316 3 88 color color NOUN cana-3316 3 89 class class NOUN cana-3316 3 90 . . PUNCT cana-3316 4 1 the the DET cana-3316 4 2 power power NOUN cana-3316 4 3 dominator dominator NOUN cana-3316 4 4 chromatic chromatic ADJ cana-3316 4 5 number number NOUN cana-3316 4 6 of of ADP cana-3316 4 7 𝐺 𝐺 PROPN cana-3316 4 8 is be AUX cana-3316 4 9 represented represent VERB cana-3316 4 10 by by ADP cana-3316 4 11 πœ’π‘π‘‘(𝐺 πœ’π‘π‘‘(𝐺 NOUN cana-3316 4 12 ) ) PUNCT cana-3316 4 13 . . PUNCT cana-3316 5 1 a a DET cana-3316 5 2 graph graph NOUN cana-3316 5 3 𝐺 𝐺 NOUN cana-3316 5 4 is be AUX cana-3316 5 5 said say VERB cana-3316 5 6 to to PART cana-3316 5 7 be be AUX cana-3316 5 8 equitably equitably ADV cana-3316 5 9 π‘˜-colorable π‘˜-colorable ADJ cana-3316 5 10 if if SCONJ cana-3316 5 11 it it PRON cana-3316 5 12 can can AUX cana-3316 5 13 be be AUX cana-3316 5 14 properly properly ADV cana-3316 5 15 colored color VERB cana-3316 5 16 with with ADP cana-3316 5 17 π‘˜ π‘˜ ADP cana-3316 5 18 colors color NOUN cana-3316 5 19 such such ADJ cana-3316 5 20 that that SCONJ cana-3316 5 21 the the DET cana-3316 5 22 size size NOUN cana-3316 5 23 of of ADP cana-3316 5 24 any any DET cana-3316 5 25 two two NUM cana-3316 5 26 color color NOUN cana-3316 5 27 classes class NOUN cana-3316 5 28 𝐢1 𝐢1 PRON cana-3316 5 29 , , PUNCT cana-3316 5 30 𝐢2 𝐢2 ADJ cana-3316 5 31 , , PUNCT cana-3316 5 32 … … PUNCT cana-3316 5 33 , , PUNCT cana-3316 5 34 πΆπ‘˜ πΆπ‘˜ PROPN cana-3316 5 35 of of ADP cana-3316 5 36 𝐺 𝐺 PROPN cana-3316 5 37 is be AUX cana-3316 5 38 differ differ ADJ cana-3316 5 39 by by ADP cana-3316 5 40 at at ADP cana-3316 5 41 most most ADV cana-3316 5 42 one one NUM cana-3316 5 43 , , PUNCT cana-3316 5 44 i.e i.e PRON cana-3316 5 45 , , PUNCT cana-3316 5 46 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 5 47 βˆ’ βˆ’ PROPN cana-3316 5 48 |𝐢𝑗|| |𝐢𝑗|| ADV cana-3316 5 49 ≀ ≀ NUM cana-3316 5 50 1,1 1,1 NUM cana-3316 5 51 ≀ ≀ NUM cana-3316 5 52 𝑖 𝑖 ADP cana-3316 5 53 , , PUNCT cana-3316 5 54 𝑗 𝑗 PROPN cana-3316 5 55 ≀ ≀ NOUN cana-3316 5 56 π‘˜ π‘˜ PRON cana-3316 5 57 and and CCONJ cana-3316 5 58 πœ’π‘’(𝐺 πœ’π‘’(𝐺 NUM cana-3316 5 59 ) ) PUNCT cana-3316 5 60 represents represent VERB cana-3316 5 61 an an DET cana-3316 5 62 equitable equitable ADJ cana-3316 5 63 chromatic chromatic ADJ cana-3316 5 64 number number NOUN cana-3316 5 65 of of ADP cana-3316 5 66 𝐺. 𝐺. NOUN cana-3316 5 67 the the DET cana-3316 5 68 power power NOUN cana-3316 5 69 dominator dominator NOUN cana-3316 5 70 equitable equitable ADJ cana-3316 5 71 coloring coloring NOUN cana-3316 5 72 of of ADP cana-3316 5 73 a a DET cana-3316 5 74 graph graph NOUN cana-3316 5 75 𝐺 𝐺 NOUN cana-3316 5 76 is be AUX cana-3316 5 77 a a DET cana-3316 5 78 proper proper ADJ cana-3316 5 79 π‘˜ π‘˜ X cana-3316 5 80 colorable colorable ADJ cana-3316 5 81 if if SCONJ cana-3316 5 82 each each DET cana-3316 5 83 vertex vertex NOUN cana-3316 5 84 of of ADP cana-3316 5 85 𝐺 𝐺 PROPN cana-3316 5 86 power power NOUN cana-3316 5 87 dominates dominate VERB cana-3316 5 88 each each DET cana-3316 5 89 and and CCONJ cana-3316 5 90 every every PRON cana-3316 5 91 vertex vertex NOUN cana-3316 5 92 of of ADP cana-3316 5 93 some some DET cana-3316 5 94 color color NOUN cana-3316 5 95 class class NOUN cana-3316 5 96 𝐢1 𝐢1 NOUN cana-3316 5 97 , , PUNCT cana-3316 5 98 𝐢2 𝐢2 ADJ cana-3316 5 99 , , PUNCT cana-3316 5 100 … … PUNCT cana-3316 5 101 πΆπ‘˜ πΆπ‘˜ VERB cana-3316 5 102 for for ADP cana-3316 5 103 which which PRON cana-3316 5 104 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 5 105 βˆ’ βˆ’ PROPN cana-3316 5 106 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 5 107 ≀ ≀ NUM cana-3316 5 108 1,1 1,1 NUM cana-3316 5 109 ≀ ≀ NUM cana-3316 5 110 𝑖 𝑖 ADP cana-3316 5 111 , , PUNCT cana-3316 5 112 𝑗 𝑗 PROPN cana-3316 5 113 ≀ ≀ NUM cana-3316 5 114 π‘˜. π‘˜. NOUN cana-3316 5 115 in in ADP cana-3316 5 116 this this DET cana-3316 5 117 paper paper NOUN cana-3316 5 118 , , PUNCT cana-3316 5 119 we we PRON cana-3316 5 120 obtain obtain VERB cana-3316 5 121 the the DET cana-3316 5 122 power power NOUN cana-3316 5 123 dominator dominator NOUN cana-3316 5 124 equitable equitable ADJ cana-3316 5 125 chromatic chromatic ADJ cana-3316 5 126 number number NOUN cana-3316 5 127 πœ’π‘π‘‘π‘’ πœ’π‘π‘‘π‘’ NOUN cana-3316 5 128 for for ADP cana-3316 5 129 some some DET cana-3316 5 130 standard standard ADJ cana-3316 5 131 graphs graph NOUN cana-3316 5 132 . . PUNCT cana-3316 6 1 keywords keyword NOUN cana-3316 6 2 : : PUNCT cana-3316 6 3 proper proper ADJ cana-3316 6 4 coloring coloring NOUN cana-3316 6 5 , , PUNCT cana-3316 6 6 color color NOUN cana-3316 6 7 class class NOUN cana-3316 6 8 , , PUNCT cana-3316 6 9 equitable equitable ADJ cana-3316 6 10 coloring coloring NOUN cana-3316 6 11 , , PUNCT cana-3316 6 12 power power NOUN cana-3316 6 13 dominator dominator NOUN cana-3316 6 14 coloring coloring NOUN cana-3316 6 15 , , PUNCT cana-3316 6 16 standard standard ADJ cana-3316 6 17 graphs graph NOUN cana-3316 6 18 . . PUNCT cana-3316 7 1 ams am NOUN cana-3316 7 2 subject subject ADJ cana-3316 7 3 classification classification NOUN cana-3316 7 4 : : PUNCT cana-3316 7 5 05c15 05c15 NOUN cana-3316 7 6 , , PUNCT cana-3316 7 7 05c69 05c69 NOUN cana-3316 7 8 1 1 NUM cana-3316 7 9 . . X cana-3316 7 10 introduction introduction NOUN cana-3316 7 11 in in ADP cana-3316 7 12 graph graph NOUN cana-3316 7 13 theory theory NOUN cana-3316 7 14 , , PUNCT cana-3316 7 15 haynes hayne NOUN cana-3316 7 16 introduced introduce VERB cana-3316 7 17 the the DET cana-3316 7 18 important important ADJ cana-3316 7 19 and and CCONJ cana-3316 7 20 extensively extensively ADV cana-3316 7 21 studied study VERB cana-3316 7 22 idea idea NOUN cana-3316 7 23 of of ADP cana-3316 7 24 domination domination NOUN cana-3316 7 25 in in ADP cana-3316 7 26 graphs graph NOUN cana-3316 7 27 [ [ X cana-3316 7 28 8 8 NUM cana-3316 7 29 ] ] PUNCT cana-3316 7 30 . . PUNCT cana-3316 8 1 in in ADP cana-3316 8 2 a a DET cana-3316 8 3 graph graph NOUN cana-3316 8 4 𝐺 𝐺 NOUN cana-3316 8 5 , , PUNCT cana-3316 8 6 a a DET cana-3316 8 7 dominating dominating NOUN cana-3316 8 8 set set NOUN cana-3316 8 9 is be AUX cana-3316 8 10 a a DET cana-3316 8 11 subset subset ADJ cana-3316 8 12 𝑆 𝑆 PROPN cana-3316 8 13 of of ADP cana-3316 8 14 its its PRON cana-3316 8 15 vertices vertex NOUN cana-3316 8 16 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 8 17 ) ) PUNCT cana-3316 8 18 , , PUNCT cana-3316 8 19 where where SCONJ cana-3316 8 20 each each DET cana-3316 8 21 vertex vertex NOUN cana-3316 8 22 outside outside ADP cana-3316 8 23 of of ADP cana-3316 8 24 𝑆 𝑆 PROPN cana-3316 8 25 shares share NOUN cana-3316 8 26 at at ADV cana-3316 8 27 least least ADV cana-3316 8 28 one one NUM cana-3316 8 29 adjacency adjacency NOUN cana-3316 8 30 with with ADP cana-3316 8 31 a a DET cana-3316 8 32 vertex vertex NOUN cana-3316 8 33 inside inside ADP cana-3316 8 34 𝑆. 𝑆. PROPN cana-3316 8 35 the the DET cana-3316 8 36 smallest small ADJ cana-3316 8 37 size size NOUN cana-3316 8 38 among among ADP cana-3316 8 39 all all DET cana-3316 8 40 such such ADJ cana-3316 8 41 dominating dominating NOUN cana-3316 8 42 sets set NOUN cana-3316 8 43 is be AUX cana-3316 8 44 termed term VERB cana-3316 8 45 as as ADP cana-3316 8 46 the the DET cana-3316 8 47 β€œ " PUNCT cana-3316 8 48 domination domination NOUN cana-3316 8 49 number number NOUN cana-3316 8 50 ” " PUNCT cana-3316 8 51 , , PUNCT cana-3316 8 52 represented represent VERB cana-3316 8 53 by by ADP cana-3316 8 54 𝛾(𝐺 𝛾(𝐺 PROPN cana-3316 8 55 ) ) PUNCT cana-3316 8 56 . . PUNCT cana-3316 9 1 𝑆 𝑆 PROPN cana-3316 9 2 is be AUX cana-3316 9 3 designated designate VERB cana-3316 9 4 as as ADP cana-3316 9 5 a a DET cana-3316 9 6 𝛾-set 𝛾-set NOUN cana-3316 9 7 of of ADP cana-3316 9 8 𝐺 𝐺 PROPN cana-3316 9 9 if if SCONJ cana-3316 9 10 it it PRON cana-3316 9 11 attains attain VERB cana-3316 9 12 this this DET cana-3316 9 13 minimal minimal ADJ cana-3316 9 14 cardinality cardinality NOUN cana-3316 10 1 [ [ X cana-3316 10 2 5 5 NUM cana-3316 10 3 ] ] PUNCT cana-3316 10 4 . . PUNCT cana-3316 11 1 haynes hayne NOUN cana-3316 11 2 et et NOUN cana-3316 12 1 al al PROPN cana-3316 13 1 [ [ X cana-3316 13 2 7 7 X cana-3316 13 3 ] ] PUNCT cana-3316 13 4 introduced introduce VERB cana-3316 13 5 a a DET cana-3316 13 6 groundbreaking groundbreake VERB cana-3316 13 7 concept concept NOUN cana-3316 13 8 in in ADP cana-3316 13 9 domination domination NOUN cana-3316 13 10 known know VERB cana-3316 13 11 as as ADP cana-3316 13 12 power power NOUN cana-3316 13 13 domination domination NOUN cana-3316 13 14 . . PUNCT cana-3316 14 1 this this DET cana-3316 14 2 concept concept NOUN cana-3316 14 3 finds find VERB cana-3316 14 4 application application NOUN cana-3316 14 5 within within ADP cana-3316 14 6 the the DET cana-3316 14 7 framework framework NOUN cana-3316 14 8 of of ADP cana-3316 14 9 electric electric ADJ cana-3316 14 10 power power NOUN cana-3316 14 11 systems system NOUN cana-3316 14 12 , , PUNCT cana-3316 14 13 where where SCONJ cana-3316 14 14 a a DET cana-3316 14 15 graph graph NOUN cana-3316 14 16 𝐺 𝐺 NOUN cana-3316 14 17 represents represent VERB cana-3316 14 18 the the DET cana-3316 14 19 system system NOUN cana-3316 14 20 , , PUNCT cana-3316 14 21 with with ADP cana-3316 14 22 vertices vertex NOUN cana-3316 14 23 symbolizing symbolize VERB cana-3316 14 24 electrical electrical ADJ cana-3316 14 25 nodes node NOUN cana-3316 14 26 and and CCONJ cana-3316 14 27 edges edge NOUN cana-3316 14 28 representing represent VERB cana-3316 14 29 transmission transmission NOUN cana-3316 14 30 lines line NOUN cana-3316 14 31 between between ADP cana-3316 14 32 these these DET cana-3316 14 33 nodes node NOUN cana-3316 14 34 . . PUNCT cana-3316 15 1 finding find VERB cana-3316 15 2 the the DET cana-3316 15 3 smallest small ADJ cana-3316 15 4 possible possible ADJ cana-3316 15 5 collection collection NOUN cana-3316 15 6 of of ADP cana-3316 15 7 β€œ " PUNCT cana-3316 15 8 phasor phasor NOUN cana-3316 15 9 measurement measurement NOUN cana-3316 15 10 units unit NOUN cana-3316 15 11 ” " PUNCT cana-3316 15 12 ( ( PUNCT cana-3316 15 13 pmus pmus NOUN cana-3316 15 14 ) ) PUNCT cana-3316 15 15 required require VERB cana-3316 15 16 for for ADP cana-3316 15 17 efficient efficient ADJ cana-3316 15 18 system system NOUN cana-3316 15 19 monitoring monitoring NOUN cana-3316 15 20 is be AUX cana-3316 15 21 the the DET cana-3316 15 22 main main ADJ cana-3316 15 23 goal goal NOUN cana-3316 15 24 . . PUNCT cana-3316 16 1 a a DET cana-3316 16 2 connected connected ADJ cana-3316 16 3 graph graph NOUN cana-3316 16 4 𝐺 𝐺 PROPN cana-3316 16 5 and and CCONJ cana-3316 16 6 a a DET cana-3316 16 7 subset subset ADJ cana-3316 16 8 𝑋 𝑋 NOUN cana-3316 16 9 of of ADP cana-3316 16 10 its its PRON cana-3316 16 11 vertices vertex NOUN cana-3316 16 12 are be AUX cana-3316 16 13 considered consider VERB cana-3316 16 14 , , PUNCT cana-3316 16 15 where where SCONJ cana-3316 16 16 the the DET cana-3316 16 17 set set NOUN cana-3316 16 18 monitored monitor VERB cana-3316 16 19 by by ADP cana-3316 16 20 𝑋 𝑋 PROPN cana-3316 16 21 denoted denote VERB cana-3316 16 22 by by ADP cana-3316 16 23 𝑀(𝑋 𝑀(𝑋 ADJ cana-3316 16 24 ) ) PUNCT cana-3316 16 25 is be AUX cana-3316 16 26 defined define VERB cana-3316 16 27 as as SCONJ cana-3316 16 28 follows follow VERB cana-3316 16 29 : : PUNCT cana-3316 16 30 1 1 X cana-3316 16 31 . . X cana-3316 16 32 initialize initialize VERB cana-3316 16 33 𝑀(𝑋 𝑀(𝑋 NUM cana-3316 16 34 ) ) PUNCT cana-3316 16 35 by by ADP cana-3316 16 36 adding add VERB cana-3316 16 37 the the DET cana-3316 16 38 vertices vertex NOUN cana-3316 16 39 in in ADP cana-3316 16 40 𝑋 𝑋 NOUN cana-3316 16 41 along along ADP cana-3316 16 42 with with ADP cana-3316 16 43 their their PRON cana-3316 16 44 neighbors neighbor NOUN cana-3316 16 45 . . PUNCT cana-3316 17 1 2 2 X cana-3316 17 2 . . X cana-3316 17 3 iterate iterate NOUN cana-3316 17 4 : : PUNCT cana-3316 17 5 while while SCONJ cana-3316 17 6 βˆƒ βˆƒ PROPN cana-3316 17 7 𝑦 𝑦 PROPN cana-3316 17 8 ∈ ∈ PROPN cana-3316 17 9 𝑀(𝑋 𝑀(𝑋 NUM cana-3316 17 10 ) ) PUNCT cana-3316 17 11 such such ADJ cana-3316 17 12 that that SCONJ cana-3316 17 13 all all DET cana-3316 17 14 its its PRON cana-3316 17 15 neighbors neighbor NOUN cana-3316 17 16 except except SCONJ cana-3316 17 17 one one NUM cana-3316 17 18 , , PUNCT cana-3316 17 19 denoted denote VERB cana-3316 17 20 by by ADP cana-3316 17 21 π‘₯ π‘₯ PROPN cana-3316 17 22 , , PUNCT cana-3316 17 23 are be AUX cana-3316 17 24 already already ADV cana-3316 17 25 in in ADP cana-3316 17 26 𝑀(𝑋 𝑀(𝑋 ADJ cana-3316 17 27 ) ) PUNCT cana-3316 17 28 then then ADV cana-3316 17 29 add add VERB cana-3316 17 30 π‘₯ π‘₯ PROPN cana-3316 17 31 to to ADP cana-3316 17 32 𝑀(𝑋 𝑀(𝑋 NUM cana-3316 17 33 ) ) PUNCT cana-3316 17 34 . . PUNCT cana-3316 18 1 after after ADP cana-3316 18 2 this this DET cana-3316 18 3 process process NOUN cana-3316 18 4 , , PUNCT cana-3316 18 5 𝑀(𝑋 𝑀(𝑋 NUM cana-3316 18 6 ) ) PUNCT cana-3316 18 7 represents represent VERB cana-3316 18 8 the the DET cana-3316 18 9 set set NOUN cana-3316 18 10 monitored monitor VERB cana-3316 18 11 by by ADP cana-3316 18 12 𝑋. 𝑋. PROPN cana-3316 18 13 a a DET cana-3316 18 14 power power NOUN cana-3316 18 15 dominating dominating NOUN cana-3316 18 16 set set VERB cana-3316 18 17 𝑋 𝑋 NOUN cana-3316 18 18 of of ADP cana-3316 18 19 𝐺 𝐺 PROPN cana-3316 18 20 is be AUX cana-3316 18 21 such such ADJ cana-3316 18 22 that that SCONJ cana-3316 18 23 𝑀(𝑋 𝑀(𝑋 ADJ cana-3316 18 24 ) ) PUNCT cana-3316 18 25 covers cover VERB cana-3316 18 26 all all DET cana-3316 18 27 vertices vertex NOUN cana-3316 18 28 in in ADP cana-3316 18 29 𝐺. 𝐺. NOUN cana-3316 18 30 the the DET cana-3316 18 31 smallest small ADJ cana-3316 18 32 size size NOUN cana-3316 18 33 of of ADP cana-3316 18 34 such such DET cana-3316 18 35 a a DET cana-3316 18 36 power power NOUN cana-3316 18 37 dominating dominating NOUN cana-3316 18 38 set set NOUN cana-3316 18 39 is be AUX cana-3316 18 40 termed term VERB cana-3316 18 41 as as ADP cana-3316 18 42 the the DET cana-3316 18 43 β€œ " PUNCT cana-3316 18 44 power power NOUN cana-3316 18 45 domination domination NOUN cana-3316 18 46 number number NOUN cana-3316 18 47 ” " PUNCT cana-3316 18 48 𝛾𝑝(𝐺 𝛾𝑝(𝐺 NOUN cana-3316 18 49 ) ) PUNCT cana-3316 18 50 . . PUNCT cana-3316 19 1 vast vast ADJ cana-3316 19 2 research research NOUN cana-3316 19 3 is be AUX cana-3316 19 4 going go VERB cana-3316 19 5 in in ADP cana-3316 19 6 [ [ X cana-3316 19 7 3 3 NUM cana-3316 19 8 , , PUNCT cana-3316 19 9 7 7 NUM cana-3316 19 10 , , PUNCT cana-3316 19 11 9 9 NUM cana-3316 19 12 , , PUNCT cana-3316 19 13 13 13 NUM cana-3316 19 14 ] ] PUNCT cana-3316 19 15 . . PUNCT cana-3316 20 1 communications communication NOUN cana-3316 20 2 on on ADP cana-3316 20 3 applied apply VERB cana-3316 20 4 nonlinear nonlinear ADJ cana-3316 20 5 analysis analysis NOUN cana-3316 20 6 issn issn NOUN cana-3316 20 7 : : PUNCT cana-3316 20 8 1074 1074 NUM cana-3316 20 9 - - PUNCT cana-3316 20 10 133x 133x NUM cana-3316 20 11 vol vol NOUN cana-3316 20 12 32 32 NUM cana-3316 20 13 no no NOUN cana-3316 20 14 . . PUNCT cana-3316 21 1 6s 6s NUM cana-3316 21 2 ( ( PUNCT cana-3316 21 3 2025 2025 NUM cana-3316 21 4 ) ) PUNCT cana-3316 21 5 530 530 NUM cana-3316 21 6 https://internationalpubls.com https://internationalpubls.com X cana-3316 21 7 the the DET cana-3316 21 8 vertex vertex NOUN cana-3316 21 9 𝑑 𝑑 NOUN cana-3316 21 10 power power NOUN cana-3316 21 11 dominates dominate VERB cana-3316 21 12 the the DET cana-3316 21 13 vertices vertex NOUN cana-3316 21 14 within within ADP cana-3316 21 15 the the DET cana-3316 21 16 set set NOUN cana-3316 21 17 𝑀(𝑑 𝑀(𝑑 PROPN cana-3316 21 18 ) ) PUNCT cana-3316 21 19 if if SCONJ cana-3316 21 20 they they PRON cana-3316 21 21 satisfy satisfy VERB cana-3316 21 22 the the DET cana-3316 21 23 following follow VERB cana-3316 21 24 conditions condition NOUN cana-3316 21 25 : : PUNCT cana-3316 22 1 1 1 X cana-3316 22 2 . . X cana-3316 22 3 each each DET cana-3316 22 4 vertex vertex NOUN cana-3316 22 5 in in ADP cana-3316 22 6 𝑁(𝑑 𝑁(𝑑 NOUN cana-3316 22 7 ) ) PUNCT cana-3316 22 8 βˆͺ βˆͺ ADP cana-3316 22 9 { { PUNCT cana-3316 22 10 𝑑 𝑑 NOUN cana-3316 22 11 } } PUNCT cana-3316 22 12 is be AUX cana-3316 22 13 included include VERB cana-3316 22 14 in in ADP cana-3316 22 15 𝑀(𝑑 𝑀(𝑑 PROPN cana-3316 22 16 ) ) PUNCT cana-3316 22 17 . . PUNCT cana-3316 23 1 2 2 X cana-3316 23 2 . . X cana-3316 23 3 for for ADP cana-3316 23 4 any any DET cana-3316 23 5 vertex vertex NOUN cana-3316 23 6 𝑐 𝑐 PROPN cana-3316 23 7 ∈ ∈ PROPN cana-3316 23 8 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 23 9 ) ) PUNCT cana-3316 23 10 , , PUNCT cana-3316 23 11 𝑐 𝑐 PROPN cana-3316 23 12 is be AUX cana-3316 23 13 added add VERB cana-3316 23 14 to to ADP cana-3316 23 15 𝑀(𝑑 𝑀(𝑑 PROPN cana-3316 23 16 ) ) PUNCT cana-3316 23 17 if if SCONJ cana-3316 23 18 there there PRON cana-3316 23 19 exists exist VERB cana-3316 23 20 a a DET cana-3316 23 21 neighbor neighbor NOUN cana-3316 23 22 𝑏 𝑏 NOUN cana-3316 23 23 in in ADP cana-3316 23 24 𝑀(𝑑 𝑀(𝑑 PROPN cana-3316 23 25 ) ) PUNCT cana-3316 23 26 such such ADJ cana-3316 23 27 that that SCONJ cana-3316 23 28 all all DET cana-3316 23 29 neighbors neighbor NOUN cana-3316 23 30 of of ADP cana-3316 23 31 𝑏 𝑏 PROPN cana-3316 23 32 except except SCONJ cana-3316 23 33 𝑐 𝑐 PROPN cana-3316 23 34 are be AUX cana-3316 23 35 already already ADV cana-3316 23 36 in in ADP cana-3316 23 37 𝑀(𝑑 𝑀(𝑑 NOUN cana-3316 23 38 ) ) PUNCT cana-3316 23 39 3 3 NUM cana-3316 23 40 . . X cana-3316 23 41 step step NOUN cana-3316 23 42 2 2 NUM cana-3316 23 43 is be AUX cana-3316 23 44 repeated repeat VERB cana-3316 23 45 for for ADP cana-3316 23 46 all all DET cana-3316 23 47 vertices vertex NOUN cana-3316 23 48 in in ADP cana-3316 23 49 the the DET cana-3316 23 50 graph graph NOUN cana-3316 23 51 . . PUNCT cana-3316 24 1 the the DET cana-3316 24 2 process process NOUN cana-3316 24 3 of of ADP cana-3316 24 4 giving give VERB cana-3316 24 5 colors color NOUN cana-3316 24 6 to to PART cana-3316 24 7 vertices vertex NOUN cana-3316 24 8 in in ADP cana-3316 24 9 a a DET cana-3316 24 10 graph graph NOUN cana-3316 24 11 𝐺 𝐺 NOUN cana-3316 24 12 so so SCONJ cana-3316 24 13 that that SCONJ cana-3316 24 14 no no DET cana-3316 24 15 two two NUM cana-3316 24 16 adjacent adjacent ADJ cana-3316 24 17 vertices vertex NOUN cana-3316 24 18 have have VERB cana-3316 24 19 the the DET cana-3316 24 20 same same ADJ cana-3316 24 21 color color NOUN cana-3316 24 22 in in ADP cana-3316 24 23 a a DET cana-3316 24 24 suitable suitable ADJ cana-3316 24 25 coloring coloring NOUN cana-3316 24 26 of of ADP cana-3316 24 27 the the DET cana-3316 24 28 graph graph NOUN cana-3316 24 29 is be AUX cana-3316 24 30 known know VERB cana-3316 24 31 as as ADP cana-3316 24 32 graph graph NOUN cana-3316 24 33 coloring coloring NOUN cana-3316 24 34 , , PUNCT cana-3316 24 35 and and CCONJ cana-3316 24 36 it it PRON cana-3316 24 37 is be AUX cana-3316 24 38 useful useful ADJ cana-3316 24 39 in in ADP cana-3316 24 40 many many ADJ cana-3316 24 41 graph graph NOUN cana-3316 24 42 theory theory NOUN cana-3316 24 43 applications application NOUN cana-3316 24 44 . . PUNCT cana-3316 25 1 let let VERB cana-3316 25 2 𝐢𝑖 𝐢𝑖 PROPN cana-3316 25 3 be be AUX cana-3316 25 4 the the DET cana-3316 25 5 color color NOUN cana-3316 25 6 class class NOUN cana-3316 25 7 𝑖 𝑖 PROPN cana-3316 25 8 , , PUNCT cana-3316 25 9 signifying signify VERB cana-3316 25 10 the the DET cana-3316 25 11 collection collection NOUN cana-3316 25 12 of of ADP cana-3316 25 13 all all DET cana-3316 25 14 vertices vertex NOUN cana-3316 25 15 that that PRON cana-3316 25 16 possess possess VERB cana-3316 25 17 the the DET cana-3316 25 18 color color NOUN cana-3316 25 19 𝑖 𝑖 PUNCT cana-3316 26 1 [ [ X cana-3316 26 2 12 12 NUM cana-3316 26 3 ] ] PUNCT cana-3316 26 4 . . PUNCT cana-3316 27 1 each each DET cana-3316 27 2 vertex vertex NOUN cana-3316 27 3 in in ADP cana-3316 27 4 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 27 5 ) ) PUNCT cana-3316 27 6 dominates dominate VERB cana-3316 27 7 every every DET cana-3316 27 8 vertex vertex NOUN cana-3316 27 9 of of ADP cana-3316 27 10 some some DET cana-3316 27 11 color color NOUN cana-3316 27 12 class class NOUN cana-3316 27 13 is be AUX cana-3316 27 14 said say VERB cana-3316 27 15 to to PART cana-3316 27 16 be be AUX cana-3316 27 17 a a DET cana-3316 27 18 dominator dominator NOUN cana-3316 27 19 coloring coloring NOUN cana-3316 27 20 of of ADP cana-3316 27 21 graph graph NOUN cana-3316 27 22 𝐺 𝐺 PROPN cana-3316 27 23 and and CCONJ cana-3316 27 24 πœ’π‘‘(𝐺 πœ’π‘‘(𝐺 PROPN cana-3316 27 25 ) ) PUNCT cana-3316 27 26 represents represent VERB cana-3316 27 27 the the DET cana-3316 27 28 dominator dominator NOUN cana-3316 27 29 chromatic chromatic ADJ cana-3316 27 30 number number NOUN cana-3316 27 31 of of ADP cana-3316 27 32 𝐺 𝐺 PROPN cana-3316 28 1 [ [ X cana-3316 28 2 10 10 NUM cana-3316 28 3 , , PUNCT cana-3316 28 4 2 2 NUM cana-3316 28 5 , , PUNCT cana-3316 28 6 12 12 NUM cana-3316 28 7 ] ] PUNCT cana-3316 28 8 . . PUNCT cana-3316 29 1 a a DET cana-3316 29 2 β€œ " PUNCT cana-3316 29 3 power power NOUN cana-3316 29 4 dominator dominator NOUN cana-3316 29 5 coloring coloring NOUN cana-3316 29 6 ” " PUNCT cana-3316 29 7 ( ( PUNCT cana-3316 29 8 pdc pdc PROPN cana-3316 29 9 ) ) PUNCT cana-3316 29 10 of of ADP cana-3316 29 11 a a DET cana-3316 29 12 graph graph NOUN cana-3316 29 13 𝐺 𝐺 NOUN cana-3316 29 14 entails entail VERB cana-3316 29 15 a a DET cana-3316 29 16 proper proper ADJ cana-3316 29 17 coloring coloring NOUN cana-3316 29 18 where where SCONJ cana-3316 29 19 each each DET cana-3316 29 20 vertex vertex NOUN cana-3316 29 21 𝑣 𝑣 ADP cana-3316 29 22 ∈ ∈ PROPN cana-3316 29 23 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 29 24 ) ) PUNCT cana-3316 29 25 power power NOUN cana-3316 29 26 dominates dominate VERB cana-3316 29 27 all all DET cana-3316 29 28 the the DET cana-3316 29 29 vertices vertex NOUN cana-3316 29 30 of of ADP cana-3316 29 31 at at ADV cana-3316 29 32 least least ADV cana-3316 29 33 one one NUM cana-3316 29 34 - - PUNCT cana-3316 29 35 color color NOUN cana-3316 29 36 class class NOUN cana-3316 29 37 . . PUNCT cana-3316 30 1 the the DET cana-3316 30 2 term term NOUN cana-3316 30 3 πœ’π‘π‘‘(𝐺 πœ’π‘π‘‘(𝐺 NUM cana-3316 30 4 ) ) PUNCT cana-3316 30 5 signifies signify VERB cana-3316 30 6 the the DET cana-3316 30 7 β€œ " PUNCT cana-3316 30 8 power power NOUN cana-3316 30 9 dominator dominator NOUN cana-3316 30 10 chromatic chromatic ADJ cana-3316 30 11 number number NOUN cana-3316 30 12 of of ADP cana-3316 30 13 𝐺 𝐺 NOUN cana-3316 30 14 ” " PUNCT cana-3316 30 15 , , PUNCT cana-3316 30 16 as as SCONJ cana-3316 30 17 elucidated elucidate VERB cana-3316 30 18 in in ADP cana-3316 30 19 reference reference NOUN cana-3316 30 20 [ [ X cana-3316 30 21 1 1 NUM cana-3316 30 22 , , PUNCT cana-3316 30 23 10 10 NUM cana-3316 30 24 ] ] PUNCT cana-3316 30 25 . . PUNCT cana-3316 31 1 a a DET cana-3316 31 2 proper proper ADJ cana-3316 31 3 coloring coloring NOUN cana-3316 31 4 of of ADP cana-3316 31 5 a a DET cana-3316 31 6 graph graph NOUN cana-3316 31 7 g g NOUN cana-3316 31 8 is be AUX cana-3316 31 9 said say VERB cana-3316 31 10 to to PART cana-3316 31 11 be be AUX cana-3316 31 12 equitably equitably ADV cana-3316 31 13 π‘˜-colorable π‘˜-colorable ADJ cana-3316 31 14 if if SCONJ cana-3316 31 15 the the DET cana-3316 31 16 number number NOUN cana-3316 31 17 of of ADP cana-3316 31 18 vertices vertex NOUN cana-3316 31 19 of of ADP cana-3316 31 20 any any DET cana-3316 31 21 two two NUM cana-3316 31 22 - - PUNCT cana-3316 31 23 color color NOUN cana-3316 31 24 classes class NOUN cana-3316 31 25 𝐢1 𝐢1 PRON cana-3316 31 26 , , PUNCT cana-3316 31 27 𝐢2 𝐢2 ADJ cana-3316 31 28 , , PUNCT cana-3316 31 29 … … PUNCT cana-3316 31 30 πΆπ‘˜ πΆπ‘˜ PROPN cana-3316 31 31 of of ADP cana-3316 31 32 𝐺 𝐺 PROPN cana-3316 31 33 is be AUX cana-3316 31 34 differ differ ADJ cana-3316 31 35 by by ADP cana-3316 31 36 at at ADP cana-3316 31 37 most most ADJ cana-3316 31 38 one one NUM cana-3316 31 39 . . PUNCT cana-3316 32 1 that that PRON cana-3316 32 2 is be AUX cana-3316 32 3 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 32 4 βˆ’ βˆ’ PROPN cana-3316 32 5 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 32 6 ≀ ≀ NUM cana-3316 32 7 1,1 1,1 NUM cana-3316 32 8 ≀ ≀ NUM cana-3316 32 9 𝑖 𝑖 ADP cana-3316 32 10 , , PUNCT cana-3316 32 11 𝑗 𝑗 PROPN cana-3316 32 12 ≀ ≀ PUNCT cana-3316 32 13 π‘˜. π‘˜. NOUN cana-3316 32 14 the the DET cana-3316 32 15 term term NOUN cana-3316 32 16 πœ’π‘’(𝐺 πœ’π‘’(𝐺 NUM cana-3316 32 17 ) ) PUNCT cana-3316 32 18 represents represent VERB cana-3316 32 19 an an DET cana-3316 32 20 β€œ " PUNCT cana-3316 32 21 equitable equitable ADJ cana-3316 32 22 chromatic chromatic ADJ cana-3316 32 23 number number NOUN cana-3316 32 24 of of ADP cana-3316 32 25 𝐺 𝐺 NOUN cana-3316 32 26 ” " PUNCT cana-3316 33 1 [ [ X cana-3316 33 2 4 4 NUM cana-3316 33 3 , , PUNCT cana-3316 33 4 11 11 NUM cana-3316 33 5 ] ] PUNCT cana-3316 33 6 . . PUNCT cana-3316 34 1 the the DET cana-3316 34 2 power power NOUN cana-3316 34 3 dominator dominator NOUN cana-3316 34 4 equitable equitable ADJ cana-3316 34 5 coloring coloring NOUN cana-3316 34 6 ( ( PUNCT cana-3316 34 7 pdec pdec PROPN cana-3316 34 8 ) ) PUNCT cana-3316 34 9 of of ADP cana-3316 34 10 a a DET cana-3316 34 11 graph graph NOUN cana-3316 34 12 𝐺 𝐺 NOUN cana-3316 34 13 defines define VERB cana-3316 34 14 each each DET cana-3316 34 15 vertex vertex NOUN cana-3316 34 16 𝑣 𝑣 ADP cana-3316 34 17 ∈ ∈ PROPN cana-3316 34 18 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 34 19 ) ) PUNCT cana-3316 34 20 power power NOUN cana-3316 34 21 dominates dominate VERB cana-3316 34 22 all all DET cana-3316 34 23 the the DET cana-3316 34 24 vertices vertex NOUN cana-3316 34 25 of of ADP cana-3316 34 26 at at ADV cana-3316 34 27 least least ADV cana-3316 34 28 one one NUM cana-3316 34 29 color color NOUN cana-3316 34 30 class class NOUN cana-3316 34 31 and and CCONJ cana-3316 34 32 also also ADV cana-3316 34 33 satisfies satisfy VERB cana-3316 34 34 the the DET cana-3316 34 35 inequality inequality NOUN cana-3316 34 36 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 34 37 βˆ’ βˆ’ PROPN cana-3316 34 38 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 34 39 ≀ ≀ NUM cana-3316 34 40 1,1 1,1 NUM cana-3316 34 41 ≀ ≀ NUM cana-3316 34 42 𝑖 𝑖 ADP cana-3316 34 43 , , PUNCT cana-3316 34 44 𝑗 𝑗 PROPN cana-3316 34 45 ≀ ≀ NOUN cana-3316 34 46 π‘˜ π‘˜ ADP cana-3316 34 47 the the DET cana-3316 34 48 notation notation NOUN cana-3316 34 49 πœ’π‘π‘‘π‘’(𝐺 πœ’π‘π‘‘π‘’(𝐺 X cana-3316 34 50 ) ) PUNCT cana-3316 34 51 represents represent VERB cana-3316 34 52 the the DET cana-3316 34 53 β€œ " PUNCT cana-3316 34 54 power power NOUN cana-3316 34 55 dominator dominator NOUN cana-3316 34 56 equitable equitable ADJ cana-3316 34 57 chromatic chromatic ADJ cana-3316 34 58 number number NOUN cana-3316 34 59 of of ADP cana-3316 34 60 𝐺 𝐺 NOUN cana-3316 34 61 ” " PUNCT cana-3316 34 62 . . PUNCT cana-3316 35 1 the the DET cana-3316 35 2 power power NOUN cana-3316 35 3 dominator dominator NOUN cana-3316 35 4 equitable equitable ADJ cana-3316 35 5 chromatic chromatic ADJ cana-3316 35 6 number number NOUN cana-3316 35 7 πœ’π‘π‘‘π‘’ πœ’π‘π‘‘π‘’ NOUN cana-3316 35 8 for for ADP cana-3316 35 9 a a DET cana-3316 35 10 few few ADJ cana-3316 35 11 common common ADJ cana-3316 35 12 graphs graph NOUN cana-3316 35 13 is be AUX cana-3316 35 14 obtained obtain VERB cana-3316 35 15 in in ADP cana-3316 35 16 this this DET cana-3316 35 17 study study NOUN cana-3316 35 18 . . PUNCT cana-3316 36 1 2 2 X cana-3316 36 2 . . X cana-3316 36 3 motivation motivation VERB cana-3316 36 4 the the DET cana-3316 36 5 motivation motivation NOUN cana-3316 36 6 behind behind ADP cana-3316 36 7 this this DET cana-3316 36 8 paper paper NOUN cana-3316 36 9 lies lie VERB cana-3316 36 10 in in ADP cana-3316 36 11 addressing address VERB cana-3316 36 12 a a DET cana-3316 36 13 significant significant ADJ cana-3316 36 14 problem problem NOUN cana-3316 36 15 in in ADP cana-3316 36 16 graph graph NOUN cana-3316 36 17 theory theory NOUN cana-3316 36 18 , , PUNCT cana-3316 36 19 finding find VERB cana-3316 36 20 efficient efficient ADJ cana-3316 36 21 and and CCONJ cana-3316 36 22 balanced balanced ADJ cana-3316 36 23 colorings coloring NOUN cana-3316 36 24 for for ADP cana-3316 36 25 graphs graph NOUN cana-3316 36 26 that that PRON cana-3316 36 27 satisfy satisfy VERB cana-3316 36 28 specific specific ADJ cana-3316 36 29 domination domination NOUN cana-3316 36 30 properties property NOUN cana-3316 36 31 . . PUNCT cana-3316 37 1 however however ADV cana-3316 37 2 , , PUNCT cana-3316 37 3 in in ADP cana-3316 37 4 certain certain ADJ cana-3316 37 5 applications application NOUN cana-3316 37 6 , , PUNCT cana-3316 37 7 such such ADJ cana-3316 37 8 as as ADP cana-3316 37 9 network network NOUN cana-3316 37 10 design design NOUN cana-3316 37 11 or or CCONJ cana-3316 37 12 resource resource NOUN cana-3316 37 13 allocation allocation NOUN cana-3316 37 14 , , PUNCT cana-3316 37 15 additional additional ADJ cana-3316 37 16 constraints constraint NOUN cana-3316 37 17 need need VERB cana-3316 37 18 to to PART cana-3316 37 19 be be AUX cana-3316 37 20 considered consider VERB cana-3316 37 21 . . PUNCT cana-3316 38 1 the the DET cana-3316 38 2 concept concept NOUN cana-3316 38 3 of of ADP cana-3316 38 4 power power NOUN cana-3316 38 5 domination domination NOUN cana-3316 38 6 introduces introduce VERB cana-3316 38 7 the the DET cana-3316 38 8 idea idea NOUN cana-3316 38 9 that that SCONJ cana-3316 38 10 each each DET cana-3316 38 11 vertex vertex NOUN cana-3316 38 12 in in ADP cana-3316 38 13 a a DET cana-3316 38 14 graph graph NOUN cana-3316 38 15 should should AUX cana-3316 38 16 have have AUX cana-3316 38 17 influence influence NOUN cana-3316 38 18 over over ADP cana-3316 38 19 all all DET cana-3316 38 20 other other ADJ cana-3316 38 21 vertices vertex NOUN cana-3316 38 22 within within ADP cana-3316 38 23 its its PRON cana-3316 38 24 neighborhood neighborhood NOUN cana-3316 38 25 . . PUNCT cana-3316 39 1 this this PRON cana-3316 39 2 leads lead VERB cana-3316 39 3 to to ADP cana-3316 39 4 the the DET cana-3316 39 5 notion notion NOUN cana-3316 39 6 of of ADP cana-3316 39 7 pdc pdc PROPN cana-3316 39 8 , , PUNCT cana-3316 39 9 where where SCONJ cana-3316 39 10 every every DET cana-3316 39 11 vertex vertex NOUN cana-3316 39 12 power power NOUN cana-3316 39 13 dominates dominate VERB cana-3316 39 14 at at ADP cana-3316 39 15 least least ADJ cana-3316 39 16 one one NUM cana-3316 39 17 - - PUNCT cana-3316 39 18 color color NOUN cana-3316 39 19 class class NOUN cana-3316 39 20 . . PUNCT cana-3316 40 1 this this PRON cana-3316 40 2 ensures ensure VERB cana-3316 40 3 a a DET cana-3316 40 4 level level NOUN cana-3316 40 5 of of ADP cana-3316 40 6 connectivity connectivity NOUN cana-3316 40 7 and and CCONJ cana-3316 40 8 influence influence NOUN cana-3316 40 9 within within ADP cana-3316 40 10 the the DET cana-3316 40 11 graph graph NOUN cana-3316 40 12 . . PUNCT cana-3316 41 1 equitable equitable ADJ cana-3316 41 2 coloring coloring NOUN cana-3316 41 3 , , PUNCT cana-3316 41 4 on on ADP cana-3316 41 5 the the DET cana-3316 41 6 other other ADJ cana-3316 41 7 hand hand NOUN cana-3316 41 8 , , PUNCT cana-3316 41 9 seeks seek VERB cana-3316 41 10 to to PART cana-3316 41 11 balance balance VERB cana-3316 41 12 the the DET cana-3316 41 13 sizes size NOUN cana-3316 41 14 of of ADP cana-3316 41 15 color color NOUN cana-3316 41 16 classes class NOUN cana-3316 41 17 . . PUNCT cana-3316 42 1 this this PRON cana-3316 42 2 is be AUX cana-3316 42 3 particularly particularly ADV cana-3316 42 4 important important ADJ cana-3316 42 5 in in ADP cana-3316 42 6 scenarios scenario NOUN cana-3316 42 7 where where SCONJ cana-3316 42 8 fairness fairness NOUN cana-3316 42 9 or or CCONJ cana-3316 42 10 resource resource NOUN cana-3316 42 11 allocation allocation NOUN cana-3316 42 12 is be AUX cana-3316 42 13 a a DET cana-3316 42 14 concern concern NOUN cana-3316 42 15 . . PUNCT cana-3316 43 1 by by ADP cana-3316 43 2 combining combine VERB cana-3316 43 3 the the DET cana-3316 43 4 principles principle NOUN cana-3316 43 5 of of ADP cana-3316 43 6 power power NOUN cana-3316 43 7 dominator dominator NOUN cana-3316 43 8 coloring coloring NOUN cana-3316 43 9 and and CCONJ cana-3316 43 10 equitable equitable ADJ cana-3316 43 11 coloring coloring NOUN cana-3316 43 12 , , PUNCT cana-3316 43 13 the the DET cana-3316 43 14 paper paper NOUN cana-3316 43 15 presented present VERB cana-3316 43 16 the the DET cana-3316 43 17 concept concept NOUN cana-3316 43 18 of of ADP cana-3316 43 19 pdec pdec NOUN cana-3316 43 20 . . PUNCT cana-3316 44 1 this this DET cana-3316 44 2 approach approach NOUN cana-3316 44 3 aims aim VERB cana-3316 44 4 to to PART cana-3316 44 5 find find VERB cana-3316 44 6 coloring coloring NOUN cana-3316 44 7 where where SCONJ cana-3316 44 8 every every DET cana-3316 44 9 vertex vertex NOUN cana-3316 44 10 not not PART cana-3316 44 11 only only ADV cana-3316 44 12 dominates dominate VERB cana-3316 44 13 at at ADP cana-3316 44 14 least least ADJ cana-3316 44 15 one one NUM cana-3316 44 16 - - PUNCT cana-3316 44 17 color color NOUN cana-3316 44 18 class class NOUN cana-3316 44 19 but but CCONJ cana-3316 44 20 also also ADV cana-3316 44 21 ensures ensure VERB cana-3316 44 22 that that SCONJ cana-3316 44 23 the the DET cana-3316 44 24 sizes size NOUN cana-3316 44 25 of of ADP cana-3316 44 26 the the DET cana-3316 44 27 color color NOUN cana-3316 44 28 classes class NOUN cana-3316 44 29 are be AUX cana-3316 44 30 balanced balanced ADJ cana-3316 44 31 . . PUNCT cana-3316 45 1 the the DET cana-3316 45 2 research research NOUN cana-3316 45 3 presented present VERB cana-3316 45 4 in in ADP cana-3316 45 5 this this DET cana-3316 45 6 paper paper NOUN cana-3316 45 7 commences commence VERB cana-3316 45 8 a a DET cana-3316 45 9 study study NOUN cana-3316 45 10 on on ADP cana-3316 45 11 this this DET cana-3316 45 12 parameter parameter NOUN cana-3316 45 13 by by ADP cana-3316 45 14 exploring explore VERB cana-3316 45 15 its its PRON cana-3316 45 16 properties property NOUN cana-3316 45 17 and and CCONJ cana-3316 45 18 determining determine VERB cana-3316 45 19 the the DET cana-3316 45 20 β€œ " PUNCT cana-3316 45 21 power power NOUN cana-3316 45 22 dominator dominator NOUN cana-3316 45 23 equitable equitable ADJ cana-3316 45 24 chromatic chromatic ADJ cana-3316 45 25 number number NOUN cana-3316 45 26 ” " PUNCT cana-3316 45 27 for for ADP cana-3316 45 28 some some DET cana-3316 45 29 standard standard ADJ cana-3316 45 30 graphs graph NOUN cana-3316 45 31 . . PUNCT cana-3316 46 1 understanding understand VERB cana-3316 46 2 this this DET cana-3316 46 3 parameter parameter NOUN cana-3316 46 4 can can AUX cana-3316 46 5 have have VERB cana-3316 46 6 implications implication NOUN cana-3316 46 7 in in ADP cana-3316 46 8 various various ADJ cana-3316 46 9 real real ADJ cana-3316 46 10 - - PUNCT cana-3316 46 11 world world NOUN cana-3316 46 12 applications application NOUN cana-3316 46 13 , , PUNCT cana-3316 46 14 such such ADJ cana-3316 46 15 as as ADP cana-3316 46 16 network network NOUN cana-3316 46 17 communication communication NOUN cana-3316 46 18 , , PUNCT cana-3316 46 19 social social ADJ cana-3316 46 20 network network NOUN cana-3316 46 21 analysis analysis NOUN cana-3316 46 22 , , PUNCT cana-3316 46 23 and and CCONJ cana-3316 46 24 resource resource NOUN cana-3316 46 25 allocation allocation NOUN cana-3316 46 26 in in ADP cana-3316 46 27 distributed distribute VERB cana-3316 46 28 systems system NOUN cana-3316 46 29 . . PUNCT cana-3316 47 1 ultimately ultimately ADV cana-3316 47 2 , , PUNCT cana-3316 47 3 this this DET cana-3316 47 4 research research NOUN cana-3316 47 5 contributes contribute VERB cana-3316 47 6 to to ADP cana-3316 47 7 advancing advance VERB cana-3316 47 8 our our PRON cana-3316 47 9 understanding understanding NOUN cana-3316 47 10 of of ADP cana-3316 47 11 graph graph NOUN cana-3316 47 12 coloring color VERB cana-3316 47 13 with with ADP cana-3316 47 14 additional additional ADJ cana-3316 47 15 domination domination NOUN cana-3316 47 16 constraints constraint NOUN cana-3316 47 17 , , PUNCT cana-3316 47 18 paving pave VERB cana-3316 47 19 the the DET cana-3316 47 20 way way NOUN cana-3316 47 21 for for ADP cana-3316 47 22 more more ADV cana-3316 47 23 efficient efficient ADJ cana-3316 47 24 and and CCONJ cana-3316 47 25 equitable equitable ADJ cana-3316 47 26 solutions solution NOUN cana-3316 47 27 in in ADP cana-3316 47 28 practical practical ADJ cana-3316 47 29 scenarios scenario NOUN cana-3316 47 30 . . PUNCT cana-3316 48 1 communications communication NOUN cana-3316 48 2 on on ADP cana-3316 48 3 applied apply VERB cana-3316 48 4 nonlinear nonlinear ADJ cana-3316 48 5 analysis analysis NOUN cana-3316 48 6 issn issn NOUN cana-3316 48 7 : : PUNCT cana-3316 48 8 1074 1074 NUM cana-3316 48 9 - - PUNCT cana-3316 48 10 133x 133x NUM cana-3316 48 11 vol vol NOUN cana-3316 48 12 32 32 NUM cana-3316 48 13 no no NOUN cana-3316 48 14 . . PUNCT cana-3316 49 1 6s 6s NUM cana-3316 49 2 ( ( PUNCT cana-3316 49 3 2025 2025 NUM cana-3316 49 4 ) ) PUNCT cana-3316 49 5 531 531 NUM cana-3316 49 6 https://internationalpubls.com https://internationalpubls.com X cana-3316 49 7 3 3 NUM cana-3316 49 8 . . PUNCT cana-3316 50 1 preliminaries preliminary NOUN cana-3316 50 2 consider consider VERB cana-3316 50 3 an an DET cana-3316 50 4 undirected undirected ADJ cana-3316 50 5 , , PUNCT cana-3316 50 6 connected connect VERB cana-3316 50 7 , , PUNCT cana-3316 50 8 and and CCONJ cana-3316 50 9 simple simple ADJ cana-3316 50 10 graph graph NOUN cana-3316 50 11 𝐺 𝐺 PROPN cana-3316 50 12 = = SYM cana-3316 50 13 ( ( PUNCT cana-3316 50 14 𝑉 𝑉 PROPN cana-3316 50 15 , , PUNCT cana-3316 50 16 𝐸 𝐸 PROPN cana-3316 50 17 ) ) PUNCT cana-3316 50 18 comprising comprise VERB cana-3316 50 19 two two NUM cana-3316 50 20 non non ADJ cana-3316 50 21 - - ADJ cana-3316 50 22 empty empty ADJ cana-3316 50 23 sets set NOUN cana-3316 50 24 𝑉 𝑉 PROPN cana-3316 50 25 and and CCONJ cana-3316 50 26 𝐸 𝐸 PROPN cana-3316 50 27 , , PUNCT cana-3316 50 28 where where SCONJ cana-3316 50 29 edges edge NOUN cana-3316 50 30 of of ADP cana-3316 50 31 𝐺 𝐺 PROPN cana-3316 50 32 are be AUX cana-3316 50 33 the the DET cana-3316 50 34 elements element NOUN cana-3316 50 35 of of ADP cana-3316 50 36 𝐸 𝐸 PROPN cana-3316 50 37 , , PUNCT cana-3316 50 38 whereas whereas SCONJ cana-3316 50 39 vertices vertex NOUN cana-3316 50 40 are be AUX cana-3316 50 41 the the DET cana-3316 50 42 components component NOUN cana-3316 50 43 of of ADP cana-3316 50 44 𝑉. 𝑉. NOUN cana-3316 50 45 we we PRON cana-3316 50 46 use use VERB cana-3316 50 47 harary harary NOUN cana-3316 50 48 's 's PART cana-3316 50 49 [ [ NOUN cana-3316 50 50 6 6 NUM cana-3316 50 51 ] ] PUNCT cana-3316 50 52 graph graph NOUN cana-3316 50 53 theoretic theoretic ADJ cana-3316 50 54 notations notation NOUN cana-3316 50 55 . . PUNCT cana-3316 51 1 if if SCONJ cana-3316 51 2 a a DET cana-3316 51 3 path path NOUN cana-3316 51 4 exists exist VERB cana-3316 51 5 between between ADP cana-3316 51 6 any any DET cana-3316 51 7 two two NUM cana-3316 51 8 vertices vertex NOUN cana-3316 51 9 in in ADP cana-3316 51 10 the the DET cana-3316 51 11 graph graph NOUN cana-3316 51 12 , , PUNCT cana-3316 51 13 denoted denote VERB cana-3316 51 14 by by ADP cana-3316 51 15 𝑒 𝑒 NOUN cana-3316 51 16 and and CCONJ cana-3316 51 17 𝑣 𝑣 ADP cana-3316 51 18 , , PUNCT cana-3316 51 19 then then ADV cana-3316 51 20 the the DET cana-3316 51 21 graph graph NOUN cana-3316 51 22 is be AUX cana-3316 51 23 said say VERB cana-3316 51 24 to to PART cana-3316 51 25 be be AUX cana-3316 51 26 connected connect VERB cana-3316 51 27 . . PUNCT cana-3316 52 1 the the DET cana-3316 52 2 open open ADJ cana-3316 52 3 neighborhood neighborhood NOUN cana-3316 52 4 of of ADP cana-3316 52 5 vertex vertex NOUN cana-3316 52 6 𝑐 𝑐 PROPN cana-3316 52 7 comprises comprise VERB cana-3316 52 8 all all DET cana-3316 52 9 vertices vertex NOUN cana-3316 52 10 adjacent adjacent ADJ cana-3316 52 11 to to ADP cana-3316 52 12 𝑐 𝑐 PROPN cana-3316 52 13 , , PUNCT cana-3316 52 14 denoted denote VERB cana-3316 52 15 as as ADP cana-3316 52 16 𝑁(𝑐 𝑁(𝑐 X cana-3316 52 17 ) ) PUNCT cana-3316 52 18 . . PUNCT cana-3316 53 1 the the DET cana-3316 53 2 union union NOUN cana-3316 53 3 of of ADP cana-3316 53 4 𝑁(𝑐 𝑁(𝑐 PROPN cana-3316 53 5 ) ) PUNCT cana-3316 53 6 with with ADP cana-3316 53 7 the the DET cana-3316 53 8 vertex vertex NOUN cana-3316 53 9 𝑐 𝑐 PROPN cana-3316 53 10 itself itself PRON cana-3316 53 11 is be AUX cana-3316 53 12 known know VERB cana-3316 53 13 as as ADP cana-3316 53 14 the the DET cana-3316 53 15 closed closed ADJ cana-3316 53 16 neighborhood neighborhood NOUN cana-3316 53 17 of of ADP cana-3316 53 18 𝑐 𝑐 NOUN cana-3316 53 19 , , PUNCT cana-3316 53 20 or or CCONJ cana-3316 53 21 𝑁[𝑐 𝑁[𝑐 PROPN cana-3316 53 22 ] ] X cana-3316 53 23 . . PUNCT cana-3316 54 1 a a DET cana-3316 54 2 path path NOUN cana-3316 54 3 𝑃𝑛 𝑃𝑛 PROPN cana-3316 54 4 is be AUX cana-3316 54 5 a a DET cana-3316 54 6 sequence sequence NOUN cana-3316 54 7 of of ADP cana-3316 54 8 𝑛 𝑛 DET cana-3316 54 9 vertices vertex NOUN cana-3316 54 10 , , PUNCT cana-3316 54 11 denoted denote VERB cana-3316 54 12 as as ADP cana-3316 54 13 π‘Ž1 π‘Ž1 NOUN cana-3316 54 14 , , PUNCT cana-3316 54 15 π‘Ž2 π‘Ž2 NOUN cana-3316 54 16 , , PUNCT cana-3316 54 17 … … PUNCT cana-3316 54 18 π‘Žπ‘› π‘Žπ‘› PRON cana-3316 54 19 that that PRON cana-3316 54 20 are be AUX cana-3316 54 21 connected connect VERB cana-3316 54 22 by by ADP cana-3316 54 23 𝑛 𝑛 DET cana-3316 54 24 βˆ’ βˆ’ NUM cana-3316 54 25 1 1 NUM cana-3316 54 26 edges edge NOUN cana-3316 54 27 , , PUNCT cana-3316 54 28 ensuring ensure VERB cana-3316 54 29 that that SCONJ cana-3316 54 30 no no DET cana-3316 54 31 vertex vertex NOUN cana-3316 54 32 or or CCONJ cana-3316 54 33 edge edge NOUN cana-3316 54 34 is be AUX cana-3316 54 35 repeated repeat VERB cana-3316 54 36 within within ADP cana-3316 54 37 the the DET cana-3316 54 38 sequence sequence NOUN cana-3316 54 39 . . PUNCT cana-3316 55 1 a a DET cana-3316 55 2 cycle cycle NOUN cana-3316 55 3 𝐢𝑛 𝐢𝑛 NOUN cana-3316 55 4 is be AUX cana-3316 55 5 a a DET cana-3316 55 6 closed closed ADJ cana-3316 55 7 path path NOUN cana-3316 55 8 having have VERB cana-3316 55 9 𝑛 𝑛 DET cana-3316 55 10 edges edge NOUN cana-3316 55 11 and and CCONJ cana-3316 55 12 𝑛 𝑛 DET cana-3316 55 13 vertices vertex NOUN cana-3316 55 14 with with ADP cana-3316 55 15 first first ADJ cana-3316 55 16 and and CCONJ cana-3316 55 17 last last ADJ cana-3316 55 18 vertex vertex NOUN cana-3316 55 19 being be AUX cana-3316 55 20 the the DET cana-3316 55 21 same same ADJ cana-3316 55 22 . . PUNCT cana-3316 56 1 all all DET cana-3316 56 2 pairs pair NOUN cana-3316 56 3 of of ADP cana-3316 56 4 vertices vertex NOUN cana-3316 56 5 in in ADP cana-3316 56 6 a a DET cana-3316 56 7 complete complete ADJ cana-3316 56 8 graph graph NOUN cana-3316 56 9 of of ADP cana-3316 56 10 order order NOUN cana-3316 56 11 𝑛 𝑛 NOUN cana-3316 56 12 , , PUNCT cana-3316 56 13 represented represent VERB cana-3316 56 14 as as ADP cana-3316 56 15 𝐾𝑛 𝐾𝑛 PROPN cana-3316 56 16 , , PUNCT cana-3316 56 17 are be AUX cana-3316 56 18 adjacent adjacent ADJ cana-3316 56 19 . . PUNCT cana-3316 57 1 in in ADP cana-3316 57 2 a a DET cana-3316 57 3 bipartite bipartite PROPN cana-3316 57 4 graph graph NOUN cana-3316 57 5 πΎπ‘š,𝑛 πΎπ‘š,𝑛 PROPN cana-3316 57 6 , , PUNCT cana-3316 57 7 there there PRON cana-3316 57 8 are be VERB cana-3316 57 9 two two NUM cana-3316 57 10 separate separate ADJ cana-3316 57 11 set set NOUN cana-3316 57 12 of of ADP cana-3316 57 13 vertices vertex NOUN cana-3316 57 14 𝑀1 𝑀1 NOUN cana-3316 57 15 and and CCONJ cana-3316 57 16 𝑀2 𝑀2 PROPN cana-3316 57 17 . . PUNCT cana-3316 58 1 set set VERB cana-3316 58 2 𝑀1 𝑀1 PROPN cana-3316 58 3 contains contain VERB cana-3316 58 4 π‘š π‘š PROPN cana-3316 58 5 vertices vertex NOUN cana-3316 58 6 , , PUNCT cana-3316 58 7 while while SCONJ cana-3316 58 8 𝑀2 𝑀2 PROPN cana-3316 58 9 contains contain VERB cana-3316 58 10 𝑛 𝑛 DET cana-3316 58 11 number number NOUN cana-3316 58 12 of of ADP cana-3316 58 13 vertices vertex NOUN cana-3316 58 14 . . PUNCT cana-3316 59 1 each each DET cana-3316 59 2 vertex vertex NOUN cana-3316 59 3 in in ADP cana-3316 59 4 𝑀1 𝑀1 PROPN cana-3316 59 5 is be AUX cana-3316 59 6 exclusively exclusively ADV cana-3316 59 7 connected connect VERB cana-3316 59 8 to to ADP cana-3316 59 9 every every DET cana-3316 59 10 other other ADJ cana-3316 59 11 vertex vertex NOUN cana-3316 59 12 in in ADP cana-3316 59 13 𝑀2 𝑀2 PROPN cana-3316 59 14 , , PUNCT cana-3316 59 15 and and CCONJ cana-3316 59 16 the the DET cana-3316 59 17 same same ADJ cana-3316 59 18 holds hold VERB cana-3316 59 19 true true ADJ cana-3316 59 20 in in ADP cana-3316 59 21 reverse reverse NOUN cana-3316 59 22 . . PUNCT cana-3316 60 1 there there PRON cana-3316 60 2 are be VERB cana-3316 60 3 no no DET cana-3316 60 4 connections connection NOUN cana-3316 60 5 between between ADP cana-3316 60 6 vertices vertex NOUN cana-3316 60 7 within within ADP cana-3316 60 8 the the DET cana-3316 60 9 same same ADJ cana-3316 60 10 set set NOUN cana-3316 60 11 . . PUNCT cana-3316 61 1 a a DET cana-3316 61 2 wheel wheel NOUN cana-3316 61 3 graph graph NOUN cana-3316 61 4 π‘Šπ‘› π‘Šπ‘› PROPN cana-3316 61 5 with with ADP cana-3316 61 6 an an DET cana-3316 61 7 order order NOUN cana-3316 61 8 of of ADP cana-3316 61 9 𝑛 𝑛 PROPN cana-3316 61 10 + + NOUN cana-3316 61 11 1 1 NUM cana-3316 61 12 , , PUNCT cana-3316 61 13 is be AUX cana-3316 61 14 formed form VERB cana-3316 61 15 by by ADP cana-3316 61 16 joining join VERB cana-3316 61 17 every every DET cana-3316 61 18 vertex vertex NOUN cana-3316 61 19 in in ADP cana-3316 61 20 cycle cycle NOUN cana-3316 62 1 𝐢𝑛 𝐢𝑛 PROPN cana-3316 62 2 to to ADP cana-3316 62 3 one one NUM cana-3316 62 4 universal universal ADJ cana-3316 62 5 vertex vertex NOUN cana-3316 62 6 ( ( PUNCT cana-3316 62 7 is be AUX cana-3316 62 8 a a DET cana-3316 62 9 vertex vertex NOUN cana-3316 62 10 that that PRON cana-3316 62 11 shares share VERB cana-3316 62 12 an an DET cana-3316 62 13 edge edge NOUN cana-3316 62 14 with with ADP cana-3316 62 15 every every DET cana-3316 62 16 other other ADJ cana-3316 62 17 vertex vertex NOUN cana-3316 62 18 present present NOUN cana-3316 62 19 in in ADP cana-3316 62 20 the the DET cana-3316 62 21 graph graph NOUN cana-3316 62 22 ) ) PUNCT cana-3316 62 23 . . PUNCT cana-3316 63 1 a a DET cana-3316 63 2 helm helm NOUN cana-3316 63 3 graph graph NOUN cana-3316 63 4 𝐻𝑛 𝐻𝑛 PROPN cana-3316 63 5 is be AUX cana-3316 63 6 constructed construct VERB cana-3316 63 7 by by ADP cana-3316 63 8 adding add VERB cana-3316 63 9 a a DET cana-3316 63 10 leaf leaf NOUN cana-3316 63 11 edge edge NOUN cana-3316 63 12 to to ADP cana-3316 63 13 each each DET cana-3316 63 14 vertex vertex NOUN cana-3316 63 15 of of ADP cana-3316 63 16 an an DET cana-3316 63 17 𝑛-wheel 𝑛-wheel NOUN cana-3316 63 18 graph graph NOUN cana-3316 63 19 , , PUNCT cana-3316 63 20 which which PRON cana-3316 63 21 consists consist VERB cana-3316 63 22 of of ADP cana-3316 63 23 a a DET cana-3316 63 24 cycle cycle NOUN cana-3316 63 25 of of ADP cana-3316 63 26 𝑛 𝑛 DET cana-3316 63 27 vertices vertex NOUN cana-3316 63 28 connected connect VERB cana-3316 63 29 to to ADP cana-3316 63 30 a a DET cana-3316 63 31 single single ADJ cana-3316 63 32 central central ADJ cana-3316 63 33 vertex vertex NOUN cana-3316 63 34 . . PUNCT cana-3316 64 1 4 4 X cana-3316 64 2 . . X cana-3316 64 3 main main ADJ cana-3316 64 4 results result NOUN cana-3316 64 5 theorem theorem VERB cana-3316 64 6 4.1 4.1 NUM cana-3316 64 7 . . PUNCT cana-3316 65 1 for for ADP cana-3316 65 2 path path NOUN cana-3316 65 3 𝑃𝑛 𝑃𝑛 PROPN cana-3316 65 4 , , PUNCT cana-3316 65 5 𝑛 𝑛 DET cana-3316 65 6 β‰₯ β‰₯ NOUN cana-3316 65 7 2 2 NUM cana-3316 65 8 , , PUNCT cana-3316 65 9 πœ’π‘π‘‘π‘’(𝑃𝑛 πœ’π‘π‘‘π‘’(𝑃𝑛 NOUN cana-3316 65 10 ) ) PUNCT cana-3316 65 11 = = SYM cana-3316 66 1 2 2 X cana-3316 66 2 . . X cana-3316 66 3 proof proof NOUN cana-3316 66 4 . . PUNCT cana-3316 67 1 let let VERB cana-3316 67 2 π‘Žπ‘– π‘Žπ‘– NUM cana-3316 67 3 , , PUNCT cana-3316 67 4 1 1 NUM cana-3316 67 5 ≀ ≀ NUM cana-3316 67 6 𝑖 𝑖 SYM cana-3316 67 7 ≀ ≀ NOUN cana-3316 67 8 𝑛 𝑛 PRON cana-3316 67 9 be be AUX cana-3316 67 10 the the DET cana-3316 67 11 𝑛 𝑛 DET cana-3316 67 12 vertices vertex NOUN cana-3316 67 13 of of ADP cana-3316 67 14 path path NOUN cana-3316 67 15 𝑃𝑛 𝑃𝑛 PROPN cana-3316 67 16 and and CCONJ cana-3316 67 17 π‘Ž1π‘Ž2 π‘Ž1π‘Ž2 X cana-3316 67 18 , , PUNCT cana-3316 67 19 π‘Ž2π‘Ž3 π‘Ž2π‘Ž3 ADP cana-3316 67 20 , , PUNCT cana-3316 67 21 … … PUNCT cana-3316 67 22 , , PUNCT cana-3316 67 23 π‘Žπ‘›βˆ’1π‘Žπ‘› π‘Žπ‘›βˆ’1π‘Žπ‘› NUM cana-3316 67 24 be be VERB cana-3316 67 25 the the DET cana-3316 67 26 edges edge NOUN cana-3316 67 27 of of ADP cana-3316 67 28 𝑃𝑛. 𝑃𝑛. NOUN cana-3316 67 29 the the DET cana-3316 67 30 odd odd ADJ cana-3316 67 31 indices index NOUN cana-3316 67 32 receive receive VERB cana-3316 67 33 color color NOUN cana-3316 67 34 1 1 NUM cana-3316 67 35 i.e i.e PROPN cana-3316 67 36 , , PUNCT cana-3316 67 37 the the DET cana-3316 67 38 vertices vertex NOUN cana-3316 67 39 π‘Ž2𝑗+1 π‘Ž2𝑗+1 VERB cana-3316 67 40 , , PUNCT cana-3316 67 41 0 0 NUM cana-3316 67 42 ≀ ≀ NUM cana-3316 67 43 𝑗 𝑗 PRON cana-3316 67 44 ≀ ≀ NUM cana-3316 67 45 ⌊(𝑛 ⌊(𝑛 NOUN cana-3316 67 46 βˆ’ βˆ’ PROPN cana-3316 67 47 1)/2βŒ‹ 1)/2βŒ‹ NUM cana-3316 67 48 receives receive VERB cana-3316 67 49 color color NOUN cana-3316 67 50 1 1 NUM cana-3316 67 51 and and CCONJ cana-3316 67 52 color color NOUN cana-3316 67 53 2 2 NUM cana-3316 67 54 to to ADP cana-3316 67 55 the the DET cana-3316 67 56 vertices vertex NOUN cana-3316 67 57 with with ADP cana-3316 67 58 the the DET cana-3316 67 59 even even ADJ cana-3316 67 60 indices index NOUN cana-3316 67 61 , , PUNCT cana-3316 67 62 i.e. i.e. X cana-3316 67 63 , , PUNCT cana-3316 67 64 to to ADP cana-3316 67 65 the the DET cana-3316 67 66 vertices vertex NOUN cana-3316 67 67 π‘Ž2𝑗 π‘Ž2𝑗 PROPN cana-3316 67 68 , , PUNCT cana-3316 67 69 1 1 NUM cana-3316 67 70 ≀ ≀ NUM cana-3316 67 71 𝑗 𝑗 PRON cana-3316 67 72 ≀ ≀ NOUN cana-3316 67 73 βŒŠπ‘›/2βŒ‹. βŒŠπ‘›/2βŒ‹. NOUN cana-3316 67 74 evidently evidently ADV cana-3316 67 75 , , PUNCT cana-3316 67 76 each each DET cana-3316 67 77 vertex vertex NOUN cana-3316 67 78 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 67 79 , , PUNCT cana-3316 67 80 1 1 NUM cana-3316 67 81 ≀ ≀ NUM cana-3316 67 82 𝑖 𝑖 SYM cana-3316 67 83 ≀ ≀ PROPN cana-3316 67 84 𝑛 𝑛 NOUN cana-3316 67 85 , , PUNCT cana-3316 67 86 power power NOUN cana-3316 67 87 dominates dominate VERB cana-3316 67 88 each each DET cana-3316 67 89 and and CCONJ cana-3316 67 90 every every PRON cana-3316 67 91 vertex vertex NOUN cana-3316 67 92 of of ADP cana-3316 67 93 𝑃𝑛. 𝑃𝑛. PROPN cana-3316 67 94 thus thus ADV cana-3316 67 95 , , PUNCT cana-3316 67 96 each each DET cana-3316 67 97 vertex vertex NOUN cana-3316 67 98 of of ADP cana-3316 67 99 𝑃𝑛 𝑃𝑛 PROPN cana-3316 67 100 power power NOUN cana-3316 67 101 dominates dominate VERB cana-3316 67 102 both both CCONJ cana-3316 67 103 the the DET cana-3316 67 104 color color NOUN cana-3316 67 105 classes class NOUN cana-3316 67 106 𝐢1 𝐢1 PRON cana-3316 67 107 and and CCONJ cana-3316 67 108 𝐢2 𝐢2 ADJ cana-3316 67 109 . . PUNCT cana-3316 68 1 furthermore furthermore ADV cana-3316 68 2 , , PUNCT cana-3316 68 3 the the DET cana-3316 68 4 discrepancy discrepancy NOUN cana-3316 68 5 occurs occur VERB cana-3316 68 6 whenever whenever SCONJ cana-3316 68 7 𝑛 𝑛 PROPN cana-3316 68 8 is be AUX cana-3316 68 9 odd odd ADJ cana-3316 68 10 , , PUNCT cana-3316 68 11 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 68 12 βˆ’ βˆ’ PROPN cana-3316 68 13 |𝐢𝑗|| |𝐢𝑗|| PROPN cana-3316 68 14 = = SYM cana-3316 68 15 1 1 NUM cana-3316 68 16 and and CCONJ cana-3316 68 17 whenever whenever SCONJ cana-3316 68 18 𝑛 𝑛 PRON cana-3316 68 19 is be AUX cana-3316 68 20 even even ADV cana-3316 68 21 , , PUNCT cana-3316 68 22 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 68 23 βˆ’ βˆ’ PROPN cana-3316 68 24 |𝐢𝑗|| |𝐢𝑗|| PROPN cana-3316 68 25 = = SYM cana-3316 68 26 0 0 PROPN cana-3316 68 27 . . PUNCT cana-3316 69 1 therefore therefore ADV cana-3316 69 2 , , PUNCT cana-3316 69 3 a a DET cana-3316 69 4 minimum minimum NOUN cana-3316 69 5 of of ADP cana-3316 69 6 2 2 NUM cana-3316 69 7 colors color NOUN cana-3316 69 8 is be AUX cana-3316 69 9 necessary necessary ADJ cana-3316 69 10 for for ADP cana-3316 69 11 achieving achieve VERB cana-3316 69 12 power power NOUN cana-3316 69 13 dominator dominator NOUN cana-3316 69 14 equitable equitable ADJ cana-3316 69 15 coloring color VERB cana-3316 69 16 . . PUNCT cana-3316 70 1 thus thus ADV cana-3316 70 2 , , PUNCT cana-3316 70 3 we we PRON cana-3316 70 4 get get VERB cana-3316 70 5 the the DET cana-3316 70 6 desired desire VERB cana-3316 70 7 result result NOUN cana-3316 70 8 πœ’π‘π‘‘π‘’(𝑃𝑛 πœ’π‘π‘‘π‘’(𝑃𝑛 PROPN cana-3316 70 9 ) ) PUNCT cana-3316 71 1 = = SYM cana-3316 71 2 2 2 X cana-3316 71 3 . . X cana-3316 71 4 theorem theorem VERB cana-3316 71 5 4.2 4.2 NUM cana-3316 71 6 . . PUNCT cana-3316 72 1 for for ADP cana-3316 72 2 cycle cycle NOUN cana-3316 72 3 𝐢𝑛 𝐢𝑛 PROPN cana-3316 72 4 , , PUNCT cana-3316 72 5 𝑛 𝑛 DET cana-3316 72 6 β‰₯ β‰₯ NOUN cana-3316 72 7 3 3 NUM cana-3316 72 8 , , PUNCT cana-3316 72 9 πœ’π‘π‘‘π‘’(𝐢𝑛 πœ’π‘π‘‘π‘’(𝐢𝑛 PROPN cana-3316 72 10 ) ) PUNCT cana-3316 72 11 = = PRON cana-3316 72 12 { { PUNCT cana-3316 72 13 2 2 NUM cana-3316 72 14 , , PUNCT cana-3316 72 15 𝑖𝑓 𝑖𝑓 NOUN cana-3316 72 16 𝑛 𝑛 DET cana-3316 72 17 𝑖𝑠 𝑖𝑠 NOUN cana-3316 72 18 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ cana-3316 72 19 3 3 NUM cana-3316 72 20 , , PUNCT cana-3316 72 21 𝑖𝑓 𝑖𝑓 ADP cana-3316 72 22 𝑛 𝑛 DET cana-3316 72 23 𝑖𝑠 𝑖𝑠 NOUN cana-3316 72 24 π‘œπ‘‘π‘‘ π‘œπ‘‘π‘‘ NOUN cana-3316 72 25 proof proof NOUN cana-3316 72 26 . . PUNCT cana-3316 73 1 we we PRON cana-3316 73 2 analyze analyze VERB cana-3316 73 3 two two NUM cana-3316 73 4 separate separate ADJ cana-3316 73 5 cases case NOUN cana-3316 73 6 for for ADP cana-3316 73 7 𝐢𝑛 𝐢𝑛 PROPN cana-3316 73 8 , , PUNCT cana-3316 73 9 based base VERB cana-3316 73 10 on on ADP cana-3316 73 11 𝑛 𝑛 PROPN cana-3316 73 12 is be AUX cana-3316 73 13 even even ADV cana-3316 73 14 or or CCONJ cana-3316 73 15 odd odd ADJ cana-3316 73 16 . . PUNCT cana-3316 74 1 case case NOUN cana-3316 74 2 1 1 NUM cana-3316 74 3 : : PUNCT cana-3316 74 4 𝑛 𝑛 PRON cana-3316 74 5 is be AUX cana-3316 74 6 even even ADV cana-3316 74 7 let let VERB cana-3316 74 8 𝐢2π‘˜ 𝐢2π‘˜ PROPN cana-3316 74 9 , , PUNCT cana-3316 74 10 π‘˜ π‘˜ DET cana-3316 74 11 β‰₯ β‰₯ NUM cana-3316 74 12 2 2 NUM cana-3316 74 13 be be AUX cana-3316 74 14 an an DET cana-3316 74 15 even even ADJ cana-3316 74 16 cycle cycle NOUN cana-3316 74 17 with with ADP cana-3316 74 18 the the DET cana-3316 74 19 vertices vertex NOUN cana-3316 74 20 π‘Ž1 π‘Ž1 ADJ cana-3316 74 21 , , PUNCT cana-3316 74 22 π‘Ž2 π‘Ž2 PROPN cana-3316 74 23 , , PUNCT cana-3316 74 24 … … PUNCT cana-3316 74 25 , , PUNCT cana-3316 74 26 π‘Ž2π‘˜ π‘Ž2π‘˜ NOUN cana-3316 74 27 , , PUNCT cana-3316 74 28 π‘˜ π‘˜ DET cana-3316 74 29 β‰₯ β‰₯ NOUN cana-3316 74 30 2 2 NUM cana-3316 74 31 and and CCONJ cana-3316 74 32 let let VERB cana-3316 74 33 π‘Ž1π‘Ž2 π‘Ž1π‘Ž2 NOUN cana-3316 74 34 , , PUNCT cana-3316 74 35 π‘Ž2π‘Ž3 π‘Ž2π‘Ž3 ADP cana-3316 74 36 , , PUNCT cana-3316 74 37 … … PUNCT cana-3316 74 38 π‘Ž2π‘˜βˆ’1π‘Ž2π‘˜ π‘Ž2π‘˜βˆ’1π‘Ž2π‘˜ NOUN cana-3316 74 39 , , PUNCT cana-3316 74 40 π‘Ž2π‘˜π‘Ž1 π‘Ž2π‘˜π‘Ž1 NOUN cana-3316 74 41 be be VERB cana-3316 74 42 the the DET cana-3316 74 43 edges edge NOUN cana-3316 74 44 of of ADP cana-3316 74 45 𝐢2π‘˜. 𝐢2π‘˜. NOUN cana-3316 74 46 we we PRON cana-3316 74 47 assign assign VERB cana-3316 74 48 color color NOUN cana-3316 74 49 1 1 NUM cana-3316 74 50 to to ADP cana-3316 74 51 the the DET cana-3316 74 52 vertices vertex NOUN cana-3316 74 53 with with ADP cana-3316 74 54 the the DET cana-3316 74 55 odd odd ADJ cana-3316 74 56 indices index NOUN cana-3316 74 57 , , PUNCT cana-3316 74 58 i.e. i.e. X cana-3316 74 59 , , PUNCT cana-3316 74 60 to to ADP cana-3316 74 61 the the DET cana-3316 74 62 vertices vertex NOUN cana-3316 74 63 π‘Ž2𝑗+1 π‘Ž2𝑗+1 VERB cana-3316 74 64 , , PUNCT cana-3316 74 65 0 0 NUM cana-3316 74 66 ≀ ≀ NUM cana-3316 75 1 𝑗 𝑗 PRON cana-3316 75 2 ≀ ≀ NUM cana-3316 75 3 ⌊(2π‘˜ ⌊(2π‘˜ ADJ cana-3316 75 4 βˆ’ βˆ’ PROPN cana-3316 75 5 1)/2βŒ‹ 1)/2βŒ‹ NUM cana-3316 75 6 and and CCONJ cana-3316 75 7 color color NOUN cana-3316 75 8 2 2 NUM cana-3316 75 9 to to ADP cana-3316 75 10 the the DET cana-3316 75 11 vertices vertex NOUN cana-3316 75 12 with with ADP cana-3316 75 13 the the DET cana-3316 75 14 even even ADJ cana-3316 75 15 indices index NOUN cana-3316 75 16 , , PUNCT cana-3316 75 17 i.e. i.e. X cana-3316 75 18 , , PUNCT cana-3316 75 19 to to ADP cana-3316 75 20 the the DET cana-3316 75 21 vertices vertex NOUN cana-3316 75 22 π‘Ž2𝑗 π‘Ž2𝑗 PROPN cana-3316 75 23 , , PUNCT cana-3316 75 24 1 1 NUM cana-3316 75 25 ≀ ≀ NUM cana-3316 75 26 𝑗 𝑗 PRON cana-3316 75 27 ≀ ≀ NOUN cana-3316 75 28 ⌊(2π‘˜)/2βŒ‹. ⌊(2π‘˜)/2βŒ‹. NUM cana-3316 75 29 each each DET cana-3316 75 30 vertex vertex NOUN cana-3316 75 31 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 75 32 , , PUNCT cana-3316 75 33 1 1 NUM cana-3316 75 34 ≀ ≀ NUM cana-3316 75 35 𝑖 𝑖 SYM cana-3316 75 36 ≀ ≀ NOUN cana-3316 75 37 𝑛 𝑛 DET cana-3316 75 38 power power NOUN cana-3316 75 39 dominates dominate VERB cana-3316 75 40 all all DET cana-3316 75 41 the the DET cana-3316 75 42 communications communication NOUN cana-3316 75 43 on on ADP cana-3316 75 44 applied apply VERB cana-3316 75 45 nonlinear nonlinear ADJ cana-3316 75 46 analysis analysis NOUN cana-3316 75 47 issn issn NOUN cana-3316 75 48 : : PUNCT cana-3316 75 49 1074 1074 NUM cana-3316 75 50 - - PUNCT cana-3316 75 51 133x 133x NUM cana-3316 75 52 vol vol NOUN cana-3316 75 53 32 32 NUM cana-3316 75 54 no no NOUN cana-3316 75 55 . . PUNCT cana-3316 76 1 6s 6s NUM cana-3316 76 2 ( ( PUNCT cana-3316 76 3 2025 2025 NUM cana-3316 76 4 ) ) PUNCT cana-3316 76 5 532 532 NUM cana-3316 76 6 https://internationalpubls.com https://internationalpubls.com X cana-3316 76 7 vertices vertex NOUN cana-3316 76 8 of of ADP cana-3316 76 9 𝐢2π‘˜ 𝐢2π‘˜ PROPN cana-3316 76 10 and and CCONJ cana-3316 76 11 hence hence ADV cana-3316 76 12 each each DET cana-3316 76 13 vertex vertex NOUN cana-3316 76 14 of of ADP cana-3316 76 15 𝐢2π‘˜ 𝐢2π‘˜ PROPN cana-3316 76 16 power power NOUN cana-3316 76 17 dominates dominate VERB cana-3316 76 18 both both CCONJ cana-3316 76 19 the the DET cana-3316 76 20 color color NOUN cana-3316 76 21 classes class NOUN cana-3316 76 22 𝐢1 𝐢1 PRON cana-3316 76 23 and and CCONJ cana-3316 76 24 𝐢2 𝐢2 NOUN cana-3316 76 25 . . PUNCT cana-3316 77 1 here here ADV cana-3316 77 2 ||𝐢1| ||𝐢1| PROPN cana-3316 77 3 βˆ’ βˆ’ PROPN cana-3316 77 4 |𝐢2|| |𝐢2|| NOUN cana-3316 77 5 = = NOUN cana-3316 77 6 0 0 X cana-3316 77 7 . . PUNCT cana-3316 78 1 therefore therefore ADV cana-3316 78 2 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 78 3 βˆ’ βˆ’ PROPN cana-3316 78 4 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 78 5 ≀ ≀ NUM cana-3316 78 6 1,1 1,1 NUM cana-3316 78 7 ≀ ≀ NUM cana-3316 78 8 𝑖 𝑖 ADP cana-3316 78 9 , , PUNCT cana-3316 78 10 𝑗 𝑗 PROPN cana-3316 78 11 ≀ ≀ ADJ cana-3316 78 12 2 2 NUM cana-3316 78 13 . . PUNCT cana-3316 78 14 hence hence ADV cana-3316 78 15 πœ’π‘π‘‘π‘’(𝐢2π‘˜ πœ’π‘π‘‘π‘’(𝐢2π‘˜ NOUN cana-3316 78 16 ) ) PUNCT cana-3316 78 17 = = SYM cana-3316 78 18 2 2 X cana-3316 78 19 . . X cana-3316 78 20 case case NOUN cana-3316 78 21 2 2 NUM cana-3316 78 22 : : PUNCT cana-3316 78 23 𝑛 𝑛 PRON cana-3316 78 24 is be AUX cana-3316 78 25 odd odd ADJ cana-3316 78 26 consider consider VERB cana-3316 78 27 the the DET cana-3316 78 28 odd odd ADJ cana-3316 78 29 cycle cycle NOUN cana-3316 78 30 𝐢2π‘˜+1 𝐢2π‘˜+1 PROPN cana-3316 78 31 , , PUNCT cana-3316 78 32 π‘˜ π‘˜ DET cana-3316 78 33 β‰₯ β‰₯ NOUN cana-3316 78 34 1 1 NUM cana-3316 78 35 with with ADP cana-3316 78 36 the the DET cana-3316 78 37 vertices vertex NOUN cana-3316 78 38 π‘Ž2𝑖+1 π‘Ž2𝑖+1 PART cana-3316 78 39 , , PUNCT cana-3316 78 40 0 0 NUM cana-3316 78 41 ≀ ≀ NUM cana-3316 78 42 𝑖 𝑖 SYM cana-3316 78 43 ≀ ≀ NOUN cana-3316 78 44 π‘˜. π‘˜. NOUN cana-3316 78 45 assigning assign VERB cana-3316 78 46 color color NOUN cana-3316 78 47 1 1 NUM cana-3316 78 48 and and CCONJ cana-3316 78 49 2 2 NUM cana-3316 78 50 alternatively alternatively ADV cana-3316 78 51 results result VERB cana-3316 78 52 violation violation NOUN cana-3316 78 53 in in ADP cana-3316 78 54 assigning assign VERB cana-3316 78 55 color color NOUN cana-3316 78 56 1 1 NUM cana-3316 78 57 and and CCONJ cana-3316 78 58 2 2 NUM cana-3316 78 59 to to ADP cana-3316 78 60 π‘Žπ‘› π‘Žπ‘› PRON cana-3316 78 61 , , PUNCT cana-3316 78 62 since since SCONJ cana-3316 78 63 neighbors neighbor NOUN cana-3316 78 64 of of ADP cana-3316 78 65 vertex vertex NOUN cana-3316 78 66 π‘Žπ‘› π‘Žπ‘› PRON cana-3316 78 67 receives receive VERB cana-3316 78 68 color color NOUN cana-3316 78 69 1 1 NUM cana-3316 78 70 and and CCONJ cana-3316 78 71 2 2 NUM cana-3316 78 72 . . X cana-3316 78 73 for for ADP cana-3316 78 74 odd odd ADJ cana-3316 78 75 values value NOUN cana-3316 78 76 of of ADP cana-3316 78 77 𝑛 𝑛 PRON cana-3316 78 78 at at ADV cana-3316 78 79 least least ADV cana-3316 78 80 3 3 NUM cana-3316 78 81 colors color NOUN cana-3316 78 82 are be AUX cana-3316 78 83 required require VERB cana-3316 78 84 . . PUNCT cana-3316 79 1 assign assign VERB cana-3316 79 2 the the DET cana-3316 79 3 colors color NOUN cana-3316 79 4 1 1 NUM cana-3316 79 5 , , PUNCT cana-3316 79 6 2 2 NUM cana-3316 79 7 and and CCONJ cana-3316 79 8 3 3 NUM cana-3316 79 9 repeatedly repeatedly ADV cana-3316 79 10 for for ADP cana-3316 79 11 the the DET cana-3316 79 12 vertices vertex NOUN cana-3316 79 13 π‘Žπ‘–+1 π‘Žπ‘–+1 NOUN cana-3316 79 14 , , PUNCT cana-3316 79 15 0 0 NUM cana-3316 79 16 ≀ ≀ NUM cana-3316 79 17 𝑖 𝑖 SYM cana-3316 79 18 ≀ ≀ NOUN cana-3316 79 19 2π‘˜ 2π‘˜ NUM cana-3316 79 20 in in ADP cana-3316 79 21 the the DET cana-3316 79 22 following following ADJ cana-3316 79 23 manner manner NOUN cana-3316 79 24 , , PUNCT cana-3316 79 25 subcase subcase NOUN cana-3316 79 26 ( ( PUNCT cana-3316 79 27 i i NOUN cana-3316 79 28 ): ): PUNCT cana-3316 79 29 𝑛 𝑛 PRON cana-3316 79 30 ≑ ≑ PROPN cana-3316 79 31 0,2 0,2 NUM cana-3316 79 32 ( ( PUNCT cana-3316 79 33 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-3316 79 34 3 3 NUM cana-3316 79 35 ) ) PUNCT cana-3316 79 36 the the DET cana-3316 79 37 vertex vertex NOUN cana-3316 79 38 set set NOUN cana-3316 79 39 { { PUNCT cana-3316 79 40 π‘Ž3π‘˜βˆ’2 π‘Ž3π‘˜βˆ’2 PROPN cana-3316 79 41 : : PUNCT cana-3316 79 42 1 1 NUM cana-3316 79 43 ≀ ≀ NOUN cana-3316 79 44 π‘˜ π‘˜ DET cana-3316 79 45 ≀ ≀ NUM cana-3316 79 46 𝑛 𝑛 DET cana-3316 79 47 3 3 NUM cana-3316 79 48 } } PUNCT cana-3316 79 49 receives receive VERB cana-3316 79 50 color color NOUN cana-3316 79 51 1 1 NUM cana-3316 79 52 . . PUNCT cana-3316 80 1 the the DET cana-3316 80 2 vertex vertex NOUN cana-3316 80 3 set set NOUN cana-3316 80 4 { { PUNCT cana-3316 80 5 π‘Ž3π‘˜βˆ’1 π‘Ž3π‘˜βˆ’1 PROPN cana-3316 80 6 : : PUNCT cana-3316 80 7 1 1 NUM cana-3316 80 8 ≀ ≀ NOUN cana-3316 80 9 π‘˜ π‘˜ DET cana-3316 80 10 ≀ ≀ NUM cana-3316 80 11 𝑛 𝑛 DET cana-3316 80 12 3 3 NUM cana-3316 80 13 } } PUNCT cana-3316 80 14 receives receive VERB cana-3316 80 15 color color NOUN cana-3316 80 16 2 2 NUM cana-3316 80 17 . . PUNCT cana-3316 81 1 the the DET cana-3316 81 2 vertices vertex NOUN cana-3316 81 3 { { PUNCT cana-3316 81 4 π‘Ž3π‘˜ π‘Ž3π‘˜ NOUN cana-3316 81 5 : : PUNCT cana-3316 81 6 1 1 NUM cana-3316 81 7 ≀ ≀ NOUN cana-3316 81 8 π‘˜ π‘˜ DET cana-3316 81 9 ≀ ≀ NUM cana-3316 81 10 𝑛 𝑛 DET cana-3316 81 11 3 3 NUM cana-3316 81 12 } } PUNCT cana-3316 81 13 receives receive VERB cana-3316 81 14 color color NOUN cana-3316 81 15 3 3 NUM cana-3316 81 16 . . PUNCT cana-3316 81 17 subcase subcase PROPN cana-3316 81 18 ( ( PUNCT cana-3316 81 19 ii ii PROPN cana-3316 81 20 ): ): PUNCT cana-3316 82 1 𝑛 𝑛 PROPN cana-3316 82 2 ≑ ≑ PROPN cana-3316 82 3 1 1 NUM cana-3316 82 4 ( ( PUNCT cana-3316 82 5 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-3316 82 6 3 3 NUM cana-3316 82 7 ) ) PUNCT cana-3316 82 8 the the DET cana-3316 82 9 vertex vertex NOUN cana-3316 82 10 set set NOUN cana-3316 82 11 { { PUNCT cana-3316 82 12 π‘Ž3π‘˜βˆ’2 π‘Ž3π‘˜βˆ’2 PROPN cana-3316 82 13 : : PUNCT cana-3316 82 14 1 1 NUM cana-3316 82 15 ≀ ≀ NOUN cana-3316 82 16 π‘˜ π‘˜ DET cana-3316 82 17 ≀ ≀ NUM cana-3316 82 18 π‘›βˆ’1 π‘›βˆ’1 NUM cana-3316 82 19 3 3 NUM cana-3316 82 20 } } PUNCT cana-3316 82 21 receives receive VERB cana-3316 82 22 color color NOUN cana-3316 82 23 1 1 NUM cana-3316 82 24 . . PUNCT cana-3316 83 1 the the DET cana-3316 83 2 vertex vertex NOUN cana-3316 83 3 set set NOUN cana-3316 83 4 { { PUNCT cana-3316 83 5 π‘Ž3π‘˜βˆ’1 π‘Ž3π‘˜βˆ’1 PROPN cana-3316 83 6 : : PUNCT cana-3316 83 7 1 1 NUM cana-3316 83 8 ≀ ≀ NOUN cana-3316 83 9 π‘˜ π‘˜ PRON cana-3316 83 10 ≀ ≀ NUM cana-3316 83 11 π‘›βˆ’1 π‘›βˆ’1 NUM cana-3316 83 12 3 3 NUM cana-3316 83 13 } } PUNCT cana-3316 83 14 βˆͺ βˆͺ NOUN cana-3316 83 15 { { PUNCT cana-3316 83 16 π‘Žπ‘› π‘Žπ‘› NOUN cana-3316 83 17 } } PUNCT cana-3316 83 18 receives receive VERB cana-3316 83 19 color color NOUN cana-3316 83 20 2 2 NUM cana-3316 83 21 . . PUNCT cana-3316 84 1 the the DET cana-3316 84 2 vertex vertex NOUN cana-3316 84 3 set set NOUN cana-3316 84 4 { { PUNCT cana-3316 84 5 π‘Ž3π‘˜ π‘Ž3π‘˜ NOUN cana-3316 84 6 : : PUNCT cana-3316 84 7 1 1 NUM cana-3316 84 8 ≀ ≀ NOUN cana-3316 84 9 π‘˜ π‘˜ DET cana-3316 84 10 ≀ ≀ NUM cana-3316 84 11 π‘›βˆ’1 π‘›βˆ’1 NUM cana-3316 84 12 3 3 NUM cana-3316 84 13 } } PUNCT cana-3316 84 14 receives receive VERB cana-3316 84 15 color color NOUN cana-3316 84 16 3 3 NUM cana-3316 84 17 . . PUNCT cana-3316 85 1 specifically specifically ADV cana-3316 85 2 , , PUNCT cana-3316 85 3 vertex vertex NOUN cana-3316 85 4 π‘Žπ‘› π‘Žπ‘› PROPN cana-3316 85 5 receives receive VERB cana-3316 85 6 color color NOUN cana-3316 85 7 2 2 NUM cana-3316 85 8 , , PUNCT cana-3316 85 9 since since SCONJ cana-3316 85 10 neighbors neighbor NOUN cana-3316 85 11 of of ADP cana-3316 85 12 π‘Žπ‘› π‘Žπ‘› PROPN cana-3316 85 13 receives receive VERB cana-3316 85 14 color color NOUN cana-3316 85 15 1 1 NUM cana-3316 85 16 and and CCONJ cana-3316 85 17 3 3 NUM cana-3316 85 18 . . X cana-3316 85 19 from from ADP cana-3316 85 20 the the DET cana-3316 85 21 above above ADJ cana-3316 85 22 subcases subcase NOUN cana-3316 85 23 , , PUNCT cana-3316 85 24 we we PRON cana-3316 85 25 clearly clearly ADV cana-3316 85 26 see see VERB cana-3316 85 27 that that SCONJ cana-3316 85 28 the the DET cana-3316 85 29 graph graph NOUN cana-3316 85 30 𝐢2π‘˜+1 𝐢2π‘˜+1 PROPN cana-3316 85 31 receives receive VERB cana-3316 85 32 proper proper ADJ cana-3316 85 33 coloring coloring NOUN cana-3316 85 34 and and CCONJ cana-3316 85 35 each each DET cana-3316 85 36 vertex vertex NOUN cana-3316 85 37 of of ADP cana-3316 85 38 𝐢2π‘˜+1 𝐢2π‘˜+1 PROPN cana-3316 85 39 power power NOUN cana-3316 85 40 dominates dominate VERB cana-3316 85 41 all all DET cana-3316 85 42 the the DET cana-3316 85 43 vertices vertex NOUN cana-3316 85 44 of of ADP cana-3316 85 45 𝐢2π‘˜+1 𝐢2π‘˜+1 PROPN cana-3316 85 46 and and CCONJ cana-3316 85 47 hence hence ADV cana-3316 85 48 each each DET cana-3316 85 49 vertex vertex NOUN cana-3316 85 50 power power NOUN cana-3316 85 51 dominates dominate VERB cana-3316 85 52 all all DET cana-3316 85 53 the the DET cana-3316 85 54 color color NOUN cana-3316 85 55 classes class NOUN cana-3316 85 56 𝐢1 𝐢1 PRON cana-3316 85 57 , , PUNCT cana-3316 85 58 𝐢2 𝐢2 ADJ cana-3316 85 59 and and CCONJ cana-3316 85 60 𝐢3 𝐢3 NOUN cana-3316 85 61 and and CCONJ cana-3316 85 62 also also ADV cana-3316 85 63 it it PRON cana-3316 85 64 satisfies satisfy VERB cana-3316 85 65 the the DET cana-3316 85 66 equitable equitable ADJ cana-3316 85 67 condition condition NOUN cana-3316 85 68 that that PRON cana-3316 85 69 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 85 70 βˆ’ βˆ’ PROPN cana-3316 85 71 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 85 72 ≀ ≀ NUM cana-3316 85 73 1,1 1,1 NUM cana-3316 85 74 ≀ ≀ NUM cana-3316 86 1 𝑖 𝑖 ADP cana-3316 86 2 , , PUNCT cana-3316 86 3 𝑗 𝑗 PROPN cana-3316 86 4 ≀ ≀ ADJ cana-3316 86 5 3 3 NUM cana-3316 86 6 . . PUNCT cana-3316 87 1 hence hence ADV cana-3316 87 2 πœ’π‘π‘‘π‘’(𝐢2π‘˜+1 πœ’π‘π‘‘π‘’(𝐢2π‘˜+1 PROPN cana-3316 87 3 ) ) PUNCT cana-3316 88 1 = = SYM cana-3316 88 2 3 3 X cana-3316 88 3 . . X cana-3316 88 4 theorem theorem VERB cana-3316 88 5 4.3 4.3 NUM cana-3316 88 6 . . PUNCT cana-3316 89 1 for for ADP cana-3316 89 2 complete complete ADJ cana-3316 89 3 graph graph NOUN cana-3316 89 4 𝐾𝑛 𝐾𝑛 PROPN cana-3316 89 5 , , PUNCT cana-3316 89 6 𝑛 𝑛 DET cana-3316 89 7 β‰₯ β‰₯ NOUN cana-3316 89 8 1 1 NUM cana-3316 89 9 , , PUNCT cana-3316 89 10 πœ’π‘π‘‘π‘’(𝐾𝑛 πœ’π‘π‘‘π‘’(𝐾𝑛 NOUN cana-3316 89 11 ) ) PUNCT cana-3316 89 12 = = SYM cana-3316 89 13 𝑛. 𝑛. NOUN cana-3316 89 14 proof proof NOUN cana-3316 89 15 . . PUNCT cana-3316 90 1 examine examine VERB cana-3316 90 2 the the DET cana-3316 90 3 entire entire ADJ cana-3316 90 4 graph graph NOUN cana-3316 90 5 𝐾𝑛 𝐾𝑛 PROPN cana-3316 90 6 , , PUNCT cana-3316 90 7 𝑛 𝑛 DET cana-3316 90 8 β‰₯ β‰₯ NOUN cana-3316 90 9 1 1 NUM cana-3316 90 10 , , PUNCT cana-3316 90 11 consisting consist VERB cana-3316 90 12 of of ADP cana-3316 90 13 the the DET cana-3316 90 14 vertices vertex NOUN cana-3316 90 15 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 90 16 , , PUNCT cana-3316 90 17 1 1 NUM cana-3316 90 18 ≀ ≀ NUM cana-3316 90 19 𝑖 𝑖 PRON cana-3316 90 20 ≀ ≀ NOUN cana-3316 90 21 𝑛. 𝑛. NOUN cana-3316 90 22 since since SCONJ cana-3316 90 23 all all DET cana-3316 90 24 the the DET cana-3316 90 25 vertices vertex NOUN cana-3316 90 26 in in ADP cana-3316 90 27 𝑉(𝐺 𝑉(𝐺 NOUN cana-3316 90 28 ) ) PUNCT cana-3316 90 29 are be AUX cana-3316 90 30 next next ADJ cana-3316 90 31 to to ADP cana-3316 90 32 each each DET cana-3316 90 33 other other ADJ cana-3316 90 34 , , PUNCT cana-3316 90 35 assign assign VERB cana-3316 90 36 color color NOUN cana-3316 90 37 𝑖 𝑖 NOUN cana-3316 90 38 , , PUNCT cana-3316 90 39 1 1 NUM cana-3316 90 40 ≀ ≀ NUM cana-3316 90 41 𝑖 𝑖 SYM cana-3316 90 42 ≀ ≀ NUM cana-3316 90 43 𝑛 𝑛 VERB cana-3316 90 44 to to ADP cana-3316 90 45 the the DET cana-3316 90 46 corresponding corresponding ADJ cana-3316 90 47 vertex vertex NOUN cana-3316 90 48 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 90 49 , , PUNCT cana-3316 90 50 1 1 NUM cana-3316 90 51 ≀ ≀ NUM cana-3316 90 52 𝑖 𝑖 SYM cana-3316 90 53 ≀ ≀ NOUN cana-3316 90 54 𝑛 𝑛 PRON cana-3316 90 55 respectively respectively ADV cana-3316 90 56 . . PUNCT cana-3316 91 1 each each DET cana-3316 91 2 vertex vertex NOUN cana-3316 91 3 in in ADP cana-3316 91 4 the the DET cana-3316 91 5 complete complete ADJ cana-3316 91 6 graph graph NOUN cana-3316 91 7 𝐾𝑛 𝐾𝑛 PROPN cana-3316 91 8 power power NOUN cana-3316 91 9 dominates dominate VERB cana-3316 91 10 every every DET cana-3316 91 11 other other ADJ cana-3316 91 12 vertex vertex NOUN cana-3316 91 13 in in ADP cana-3316 91 14 𝐾𝑛 𝐾𝑛 PROPN cana-3316 91 15 , , PUNCT cana-3316 91 16 as as ADV cana-3316 91 17 well well ADV cana-3316 91 18 as as ADP cana-3316 91 19 all all DET cana-3316 91 20 color color NOUN cana-3316 91 21 classes class NOUN cana-3316 91 22 . . PUNCT cana-3316 92 1 furthermore furthermore ADV cana-3316 92 2 , , PUNCT cana-3316 92 3 the the DET cana-3316 92 4 inequality inequality NOUN cana-3316 92 5 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 92 6 βˆ’ βˆ’ PROPN cana-3316 92 7 |𝐢𝑗|| |𝐢𝑗|| PROPN cana-3316 93 1 = = SYM cana-3316 93 2 0 0 NUM cana-3316 93 3 is be AUX cana-3316 93 4 satisfied satisfied ADJ cana-3316 93 5 for for ADP cana-3316 93 6 all all DET cana-3316 93 7 pairs pair NOUN cana-3316 93 8 of of ADP cana-3316 93 9 color color NOUN cana-3316 93 10 classes class NOUN cana-3316 93 11 𝐢𝑖 𝐢𝑖 PROPN cana-3316 93 12 and and CCONJ cana-3316 93 13 𝐢𝑗 𝐢𝑗 PROPN cana-3316 93 14 , , PUNCT cana-3316 93 15 where where SCONJ cana-3316 93 16 1 1 NUM cana-3316 93 17 ≀ ≀ NUM cana-3316 93 18 𝑖 𝑖 PUNCT cana-3316 93 19 , , PUNCT cana-3316 93 20 𝑗 𝑗 NOUN cana-3316 93 21 ≀ ≀ ADJ cana-3316 93 22 𝑛. 𝑛. NOUN cana-3316 93 23 hence hence ADV cana-3316 93 24 πœ’π‘π‘‘π‘’(𝐾𝑛 πœ’π‘π‘‘π‘’(𝐾𝑛 ADJ cana-3316 93 25 ) ) PUNCT cana-3316 93 26 = = SYM cana-3316 93 27 𝑛. 𝑛. NOUN cana-3316 93 28 theorem theorem VERB cana-3316 93 29 4.4 4.4 NUM cana-3316 93 30 . . PUNCT cana-3316 94 1 for for ADP cana-3316 94 2 complete complete ADJ cana-3316 94 3 bipartite bipartite PROPN cana-3316 94 4 graph graph NOUN cana-3316 94 5 πΎπ‘š,𝑛 πΎπ‘š,𝑛 PROPN cana-3316 94 6 , , PUNCT cana-3316 94 7 π‘š π‘š PROPN cana-3316 94 8 , , PUNCT cana-3316 94 9 𝑛 𝑛 DET cana-3316 94 10 β‰₯ β‰₯ NUM cana-3316 94 11 1 1 NUM cana-3316 94 12 πœ’π‘π‘‘π‘’(πΎπ‘š,𝑛 πœ’π‘π‘‘π‘’(πΎπ‘š,𝑛 NOUN cana-3316 94 13 ) ) PUNCT cana-3316 94 14 = = PRON cana-3316 94 15 { { PUNCT cana-3316 94 16 2 2 NUM cana-3316 94 17 , , PUNCT cana-3316 94 18 𝑖𝑓 𝑖𝑓 X cana-3316 94 19 ( ( PUNCT cana-3316 94 20 π‘š π‘š PROPN cana-3316 94 21 βˆ’ βˆ’ PROPN cana-3316 94 22 𝑛 𝑛 NOUN cana-3316 94 23 ) ) PUNCT cana-3316 94 24 < < X cana-3316 94 25 2 2 NUM cana-3316 94 26 ⌈ ⌈ NOUN cana-3316 94 27 π‘š π‘š ADP cana-3316 94 28 𝑛 𝑛 PROPN cana-3316 94 29 + + NUM cana-3316 94 30 1 1 NUM cana-3316 94 31 βŒ‰ βŒ‰ NOUN cana-3316 94 32 + + NOUN cana-3316 94 33 1 1 NUM cana-3316 94 34 , , PUNCT cana-3316 94 35 𝑖𝑓 𝑖𝑓 X cana-3316 94 36 ( ( PUNCT cana-3316 94 37 π‘š π‘š PROPN cana-3316 94 38 βˆ’ βˆ’ PROPN cana-3316 94 39 𝑛 𝑛 PROPN cana-3316 94 40 ) ) PUNCT cana-3316 94 41 β‰₯ β‰₯ NOUN cana-3316 94 42 2 2 NUM cana-3316 94 43 proof proof NOUN cana-3316 94 44 . . PUNCT cana-3316 95 1 let let AUX cana-3316 95 2 ( ( PUNCT cana-3316 95 3 𝑀1 𝑀1 PROPN cana-3316 95 4 , , PUNCT cana-3316 95 5 𝑀2 𝑀2 PROPN cana-3316 95 6 ) ) PUNCT cana-3316 95 7 be be VERB cana-3316 95 8 the the DET cana-3316 95 9 partitions partition NOUN cana-3316 95 10 of of ADP cana-3316 95 11 πΎπ‘š,𝑛. πΎπ‘š,𝑛. PROPN cana-3316 95 12 let let VERB cana-3316 95 13 π‘š π‘š PART cana-3316 95 14 be be AUX cana-3316 95 15 the the DET cana-3316 95 16 cardinality cardinality NOUN cana-3316 95 17 of of ADP cana-3316 95 18 𝑀1 𝑀1 PROPN cana-3316 95 19 and and CCONJ cana-3316 95 20 𝑛 𝑛 PROPN cana-3316 95 21 be be AUX cana-3316 95 22 the the DET cana-3316 95 23 cardinality cardinality NOUN cana-3316 95 24 of of ADP cana-3316 95 25 𝑀2 𝑀2 PROPN cana-3316 95 26 . . PUNCT cana-3316 96 1 case case NOUN cana-3316 96 2 1 1 NUM cana-3316 96 3 : : PUNCT cana-3316 96 4 ( ( PUNCT cana-3316 96 5 π‘š π‘š PROPN cana-3316 96 6 βˆ’ βˆ’ PROPN cana-3316 96 7 𝑛 𝑛 NOUN cana-3316 96 8 ) ) PUNCT cana-3316 96 9 < < X cana-3316 96 10 2 2 NUM cana-3316 96 11 each each DET cana-3316 96 12 vertex vertex NOUN cana-3316 96 13 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 96 14 , , PUNCT cana-3316 96 15 1 1 NUM cana-3316 96 16 ≀ ≀ NUM cana-3316 96 17 𝑖 𝑖 SYM cana-3316 96 18 ≀ ≀ NUM cana-3316 96 19 π‘š π‘š NOUN cana-3316 96 20 of of ADP cana-3316 96 21 𝑀1 𝑀1 PROPN cana-3316 96 22 power power NOUN cana-3316 96 23 dominates dominate VERB cana-3316 96 24 𝑁[π‘Žπ‘– 𝑁[π‘Žπ‘– ADV cana-3316 96 25 ] ] X cana-3316 96 26 , , PUNCT cana-3316 96 27 1 1 NUM cana-3316 96 28 ≀ ≀ NUM cana-3316 96 29 𝑖 𝑖 SYM cana-3316 96 30 ≀ ≀ NOUN cana-3316 96 31 π‘š. π‘š. ADV cana-3316 96 32 each each DET cana-3316 96 33 vertex vertex NOUN cana-3316 96 34 π‘Žπ‘— π‘Žπ‘— ADP cana-3316 96 35 , , PUNCT cana-3316 96 36 1 1 NUM cana-3316 96 37 ≀ ≀ NUM cana-3316 96 38 𝑗 𝑗 PRON cana-3316 96 39 ≀ ≀ NUM cana-3316 96 40 𝑛 𝑛 PRON cana-3316 96 41 of of ADP cana-3316 96 42 𝑀2 𝑀2 PROPN cana-3316 96 43 power power NOUN cana-3316 96 44 dominates dominate VERB cana-3316 96 45 𝑁[π‘Žπ‘— 𝑁[π‘Žπ‘— NOUN cana-3316 96 46 ] ] X cana-3316 96 47 , , PUNCT cana-3316 96 48 1 1 NUM cana-3316 96 49 ≀ ≀ NUM cana-3316 96 50 𝑗 𝑗 PRON cana-3316 96 51 ≀ ≀ ADJ cana-3316 96 52 𝑛. 𝑛. NOUN cana-3316 96 53 by by ADP cana-3316 96 54 assigning assign VERB cana-3316 96 55 color color NOUN cana-3316 96 56 1 1 NUM cana-3316 96 57 to to ADP cana-3316 96 58 vertices vertex NOUN cana-3316 96 59 in in ADP cana-3316 96 60 𝑀1 𝑀1 NOUN cana-3316 96 61 and and CCONJ cana-3316 96 62 color color NOUN cana-3316 96 63 2 2 NUM cana-3316 96 64 to to AUX cana-3316 96 65 vertices vertex NOUN cana-3316 96 66 in in ADP cana-3316 96 67 communications communication NOUN cana-3316 96 68 on on ADP cana-3316 96 69 applied apply VERB cana-3316 96 70 nonlinear nonlinear ADJ cana-3316 96 71 analysis analysis NOUN cana-3316 96 72 issn issn NOUN cana-3316 96 73 : : PUNCT cana-3316 96 74 1074 1074 NUM cana-3316 96 75 - - PUNCT cana-3316 96 76 133x 133x NUM cana-3316 96 77 vol vol NOUN cana-3316 96 78 32 32 NUM cana-3316 96 79 no no NOUN cana-3316 96 80 . . PUNCT cana-3316 97 1 6s 6s NUM cana-3316 97 2 ( ( PUNCT cana-3316 97 3 2025 2025 NUM cana-3316 97 4 ) ) PUNCT cana-3316 97 5 533 533 NUM cana-3316 97 6 https://internationalpubls.com https://internationalpubls.com X cana-3316 97 7 𝑀2 𝑀2 PROPN cana-3316 97 8 , , PUNCT cana-3316 97 9 we we PRON cana-3316 97 10 ensure ensure VERB cana-3316 97 11 that that SCONJ cana-3316 97 12 each each DET cana-3316 97 13 vertex vertex NOUN cana-3316 97 14 in in ADP cana-3316 97 15 𝑀1 𝑀1 PROPN cana-3316 97 16 power power NOUN cana-3316 97 17 dominates dominate VERB cana-3316 97 18 the the DET cana-3316 97 19 color color NOUN cana-3316 97 20 class class NOUN cana-3316 97 21 𝐢2 𝐢2 PROPN cana-3316 97 22 , , PUNCT cana-3316 97 23 while while SCONJ cana-3316 97 24 each each DET cana-3316 97 25 vertex vertex NOUN cana-3316 97 26 in in ADP cana-3316 97 27 𝑀2 𝑀2 PROPN cana-3316 97 28 power power NOUN cana-3316 97 29 dominates dominate VERB cana-3316 97 30 the the DET cana-3316 97 31 color color NOUN cana-3316 97 32 class class NOUN cana-3316 97 33 𝐢1 𝐢1 NOUN cana-3316 97 34 . . PUNCT cana-3316 98 1 therefore therefore ADV cana-3316 98 2 ||𝐢1| ||𝐢1| PROPN cana-3316 98 3 βˆ’ βˆ’ PROPN cana-3316 98 4 |𝐢2|| |𝐢2|| NOUN cana-3316 98 5 ≀ ≀ NOUN cana-3316 98 6 1 1 NUM cana-3316 98 7 . . PUNCT cana-3316 98 8 hence hence ADV cana-3316 98 9 πœ’π‘π‘‘π‘’(πΎπ‘š,𝑛 πœ’π‘π‘‘π‘’(πΎπ‘š,𝑛 PROPN cana-3316 98 10 ) ) PUNCT cana-3316 98 11 = = SYM cana-3316 98 12 2 2 X cana-3316 98 13 . . X cana-3316 98 14 case case NOUN cana-3316 98 15 2 2 NUM cana-3316 98 16 : : PUNCT cana-3316 98 17 ( ( PUNCT cana-3316 98 18 π‘š π‘š PROPN cana-3316 98 19 βˆ’ βˆ’ PROPN cana-3316 98 20 𝑛 𝑛 NOUN cana-3316 98 21 ) ) PUNCT cana-3316 98 22 β‰₯ β‰₯ NOUN cana-3316 98 23 2 2 NUM cana-3316 98 24 without without ADP cana-3316 98 25 loss loss NOUN cana-3316 98 26 of of ADP cana-3316 98 27 generality generality NOUN cana-3316 98 28 , , PUNCT cana-3316 98 29 let let VERB cana-3316 98 30 π‘š π‘š PRON cana-3316 98 31 > > X cana-3316 98 32 𝑛. 𝑛. NOUN cana-3316 98 33 each each DET cana-3316 98 34 vertex vertex NOUN cana-3316 98 35 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 98 36 , , PUNCT cana-3316 98 37 1 1 NUM cana-3316 98 38 ≀ ≀ NUM cana-3316 98 39 𝑖 𝑖 PUNCT cana-3316 98 40 ≀ ≀ NOUN cana-3316 98 41 π‘š π‘š ADP cana-3316 98 42 , , PUNCT cana-3316 98 43 of of ADP cana-3316 98 44 partition partition NOUN cana-3316 98 45 𝑀1 𝑀1 PROPN cana-3316 98 46 power power NOUN cana-3316 98 47 dominates dominate VERB cana-3316 98 48 𝑁[π‘Žπ‘– 𝑁[π‘Žπ‘– ADV cana-3316 98 49 ] ] X cana-3316 98 50 , , PUNCT cana-3316 98 51 1 1 NUM cana-3316 98 52 ≀ ≀ NUM cana-3316 98 53 𝑖 𝑖 SYM cana-3316 98 54 ≀ ≀ NOUN cana-3316 98 55 π‘š π‘š X cana-3316 98 56 and and CCONJ cana-3316 98 57 each each DET cana-3316 98 58 vertex vertex NOUN cana-3316 98 59 π‘Žπ‘— π‘Žπ‘— ADP cana-3316 98 60 , , PUNCT cana-3316 98 61 1 1 NUM cana-3316 98 62 ≀ ≀ NUM cana-3316 98 63 𝑗 𝑗 PRON cana-3316 98 64 ≀ ≀ NUM cana-3316 98 65 𝑛 𝑛 NOUN cana-3316 98 66 , , PUNCT cana-3316 98 67 𝑛 𝑛 PROPN cana-3316 98 68 > > X cana-3316 98 69 2 2 NUM cana-3316 98 70 of of ADP cana-3316 98 71 partition partition NOUN cana-3316 98 72 𝑀2 𝑀2 PROPN cana-3316 98 73 power power NOUN cana-3316 98 74 dominates dominate VERB cana-3316 98 75 𝑁[π‘Žπ‘— 𝑁[π‘Žπ‘— NOUN cana-3316 98 76 ] ] X cana-3316 98 77 , , PUNCT cana-3316 98 78 1 1 NUM cana-3316 98 79 ≀ ≀ NUM cana-3316 98 80 𝑗 𝑗 PRON cana-3316 98 81 ≀ ≀ ADJ cana-3316 98 82 𝑛. 𝑛. NOUN cana-3316 98 83 divide divide VERB cana-3316 98 84 the the DET cana-3316 98 85 vertices vertex NOUN cana-3316 98 86 of of ADP cana-3316 98 87 𝑀1 𝑀1 NOUN cana-3316 98 88 into into ADP cana-3316 98 89 ⌈ ⌈ PROPN cana-3316 98 90 π‘š π‘š ADP cana-3316 98 91 𝑛+1 𝑛+1 PROPN cana-3316 98 92 βŒ‰ βŒ‰ X cana-3316 98 93 number number NOUN cana-3316 98 94 of of ADP cana-3316 98 95 sets set NOUN cana-3316 98 96 and and CCONJ cana-3316 98 97 utilize utilize VERB cana-3316 98 98 ⌈ ⌈ X cana-3316 98 99 π‘š π‘š ADP cana-3316 98 100 𝑛+1 𝑛+1 PROPN cana-3316 98 101 βŒ‰ βŒ‰ PRON cana-3316 98 102 colors color NOUN cana-3316 98 103 to to PART cana-3316 98 104 color color VERB cana-3316 98 105 the the DET cana-3316 98 106 vertices vertex NOUN cana-3316 98 107 in in ADP cana-3316 98 108 𝑀1 𝑀1 NOUN cana-3316 98 109 and and CCONJ cana-3316 98 110 assign assign VERB cana-3316 98 111 color color NOUN cana-3316 98 112 1 1 NUM cana-3316 98 113 to to ADP cana-3316 98 114 all all DET cana-3316 98 115 vertices vertex NOUN cana-3316 98 116 of of ADP cana-3316 98 117 𝑀2 𝑀2 PROPN cana-3316 98 118 . . PUNCT cana-3316 99 1 this this PRON cana-3316 99 2 ensures ensure VERB cana-3316 99 3 that that SCONJ cana-3316 99 4 every every DET cana-3316 99 5 vertex vertex NOUN cana-3316 99 6 power power NOUN cana-3316 99 7 dominates dominate VERB cana-3316 99 8 at at ADP cana-3316 99 9 least least ADJ cana-3316 99 10 one one NUM cana-3316 99 11 - - PUNCT cana-3316 99 12 color color NOUN cana-3316 99 13 class class NOUN cana-3316 99 14 and and CCONJ cana-3316 99 15 maintains maintain VERB cana-3316 99 16 the the DET cana-3316 99 17 equitable equitable ADJ cana-3316 99 18 condition condition NOUN cana-3316 99 19 . . PUNCT cana-3316 100 1 hence hence ADV cana-3316 100 2 πœ’π‘π‘‘π‘’(πΎπ‘š,𝑛 πœ’π‘π‘‘π‘’(πΎπ‘š,𝑛 PROPN cana-3316 100 3 ) ) PUNCT cana-3316 100 4 = = PUNCT cana-3316 101 1 ⌈ ⌈ SYM cana-3316 101 2 π‘š π‘š ADP cana-3316 101 3 𝑛+1 𝑛+1 PROPN cana-3316 101 4 βŒ‰ βŒ‰ X cana-3316 101 5 + + PROPN cana-3316 101 6 1 1 X cana-3316 101 7 . . X cana-3316 101 8 theorem theorem VERB cana-3316 101 9 4.5 4.5 NUM cana-3316 101 10 . . PUNCT cana-3316 102 1 for for ADP cana-3316 102 2 wheel wheel NOUN cana-3316 102 3 graph graph NOUN cana-3316 102 4 π‘Šπ‘› π‘Šπ‘› PROPN cana-3316 102 5 , , PUNCT cana-3316 102 6 𝑛 𝑛 DET cana-3316 102 7 β‰₯ β‰₯ NOUN cana-3316 102 8 4 4 NUM cana-3316 102 9 , , PUNCT cana-3316 102 10 πœ’π‘π‘‘π‘’(π‘Šπ‘› πœ’π‘π‘‘π‘’(π‘Šπ‘› NOUN cana-3316 102 11 ) ) PUNCT cana-3316 102 12 = = PUNCT cana-3316 103 1 ⌈ ⌈ NOUN cana-3316 103 2 𝑛 𝑛 ADP cana-3316 103 3 2 2 NUM cana-3316 103 4 βŒ‰ βŒ‰ NOUN cana-3316 103 5 + + CCONJ cana-3316 103 6 1 1 NUM cana-3316 103 7 proof proof NOUN cana-3316 103 8 . . PUNCT cana-3316 104 1 consider consider VERB cana-3316 104 2 π‘Ž1 π‘Ž1 NOUN cana-3316 104 3 as as ADP cana-3316 104 4 the the DET cana-3316 104 5 central central ADJ cana-3316 104 6 vertex vertex NOUN cana-3316 104 7 and and CCONJ cana-3316 104 8 π‘Žπ‘– π‘Žπ‘– CCONJ cana-3316 104 9 , , PUNCT cana-3316 104 10 where where SCONJ cana-3316 104 11 2 2 NUM cana-3316 104 12 ≀ ≀ NOUN cana-3316 104 13 𝑖 𝑖 SYM cana-3316 104 14 ≀ ≀ NUM cana-3316 104 15 𝑛 𝑛 PRON cana-3316 104 16 as as ADP cana-3316 104 17 the the DET cana-3316 104 18 vertices vertex NOUN cana-3316 104 19 located locate VERB cana-3316 104 20 on on ADP cana-3316 104 21 the the DET cana-3316 104 22 cycle cycle NOUN cana-3316 104 23 of of ADP cana-3316 104 24 π‘Šπ‘›. π‘Šπ‘›. PROPN cana-3316 104 25 assign assign NOUN cana-3316 104 26 color color NOUN cana-3316 104 27 1 1 NUM cana-3316 104 28 to to PART cana-3316 104 29 π‘Ž1 π‘Ž1 VERB cana-3316 104 30 . . PUNCT cana-3316 105 1 each each DET cana-3316 105 2 vertex vertex NOUN cana-3316 105 3 of of ADP cana-3316 105 4 𝑛 𝑛 DET cana-3316 105 5 wheel wheel NOUN cana-3316 105 6 graph graph NOUN cana-3316 105 7 power power NOUN cana-3316 105 8 dominates dominate VERB cana-3316 105 9 the the DET cana-3316 105 10 color color NOUN cana-3316 105 11 class class NOUN cana-3316 105 12 𝐢1 𝐢1 NOUN cana-3316 105 13 . . PUNCT cana-3316 106 1 assign assign VERB cana-3316 106 2 ⌈ ⌈ NOUN cana-3316 106 3 𝑛 𝑛 ADP cana-3316 106 4 2 2 NUM cana-3316 106 5 βŒ‰ βŒ‰ VERB cana-3316 106 6 colors color NOUN cana-3316 106 7 to to ADP cana-3316 106 8 the the DET cana-3316 106 9 remaining remain VERB cana-3316 106 10 vertices vertex NOUN cana-3316 106 11 equitably equitably ADV cana-3316 106 12 , , PUNCT cana-3316 106 13 so so SCONJ cana-3316 106 14 that that PRON cana-3316 106 15 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 106 16 βˆ’ βˆ’ PROPN cana-3316 106 17 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 106 18 ≀ ≀ NUM cana-3316 106 19 1 1 NUM cana-3316 106 20 βˆ€ βˆ€ NOUN cana-3316 106 21 𝑖 𝑖 NOUN cana-3316 106 22 and and CCONJ cana-3316 106 23 𝑗. 𝑗. NOUN cana-3316 106 24 therefore therefore ADV cana-3316 106 25 πœ’π‘π‘‘π‘’(π‘Šπ‘› πœ’π‘π‘‘π‘’(π‘Šπ‘› PROPN cana-3316 106 26 ) ) PUNCT cana-3316 107 1 = = PUNCT cana-3316 107 2 ⌈ ⌈ NOUN cana-3316 107 3 𝑛 𝑛 DET cana-3316 107 4 2 2 NUM cana-3316 107 5 βŒ‰ βŒ‰ NOUN cana-3316 107 6 + + CCONJ cana-3316 107 7 1 1 X cana-3316 107 8 . . X cana-3316 107 9 theorem theorem VERB cana-3316 107 10 4.6 4.6 NUM cana-3316 107 11 . . PUNCT cana-3316 108 1 for for ADP cana-3316 108 2 helm helm NOUN cana-3316 108 3 graph graph NOUN cana-3316 108 4 𝐻𝑛 𝐻𝑛 PROPN cana-3316 108 5 , , PUNCT cana-3316 108 6 𝑛 𝑛 DET cana-3316 108 7 β‰₯ β‰₯ NOUN cana-3316 108 8 3 3 NUM cana-3316 108 9 , , PUNCT cana-3316 108 10 πœ’π‘π‘‘π‘’(𝐻𝑛 πœ’π‘π‘‘π‘’(𝐻𝑛 PROPN cana-3316 108 11 ) ) PUNCT cana-3316 108 12 = = SYM cana-3316 108 13 𝑛 𝑛 PROPN cana-3316 109 1 + + SYM cana-3316 109 2 ⌈ ⌈ NOUN cana-3316 109 3 𝑛+1 𝑛+1 ADP cana-3316 109 4 2 2 NUM cana-3316 109 5 βŒ‰. βŒ‰. ADJ cana-3316 109 6 proof proof NOUN cana-3316 109 7 . . PUNCT cana-3316 110 1 let let VERB cana-3316 110 2 π‘Žπ‘– π‘Žπ‘– SYM cana-3316 110 3 , , PUNCT cana-3316 110 4 1 1 NUM cana-3316 110 5 ≀ ≀ NUM cana-3316 110 6 𝑖 𝑖 SYM cana-3316 110 7 ≀ ≀ NOUN cana-3316 110 8 𝑛 𝑛 PRON cana-3316 110 9 be be AUX cana-3316 110 10 the the DET cana-3316 110 11 vertices vertex NOUN cana-3316 110 12 on on ADP cana-3316 110 13 the the DET cana-3316 110 14 cycle cycle NOUN cana-3316 110 15 of of ADP cana-3316 110 16 𝐻𝑛 𝐻𝑛 PROPN cana-3316 110 17 , , PUNCT cana-3316 110 18 𝑒𝑖 𝑒𝑖 ADP cana-3316 110 19 , , PUNCT cana-3316 110 20 1 1 NUM cana-3316 110 21 ≀ ≀ NUM cana-3316 110 22 𝑖 𝑖 SYM cana-3316 110 23 ≀ ≀ NOUN cana-3316 110 24 𝑛 𝑛 ADP cana-3316 110 25 representing represent VERB cana-3316 110 26 the the DET cana-3316 110 27 pendent pendent ADJ cana-3316 110 28 vertices vertex NOUN cana-3316 110 29 and and CCONJ cana-3316 110 30 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-3316 110 31 denotes denote NOUN cana-3316 110 32 the the DET cana-3316 110 33 central central ADJ cana-3316 110 34 vertex vertex NOUN cana-3316 110 35 within within ADP cana-3316 110 36 the the DET cana-3316 110 37 graph graph NOUN cana-3316 110 38 𝐻𝑛. 𝐻𝑛. VERB cana-3316 110 39 the the DET cana-3316 110 40 vertex vertex NOUN cana-3316 110 41 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-3316 110 42 power power NOUN cana-3316 110 43 dominates dominate VERB cana-3316 110 44 over over ADP cana-3316 110 45 all all DET cana-3316 110 46 vertices vertex NOUN cana-3316 110 47 of of ADP cana-3316 110 48 𝐻𝑛. 𝐻𝑛. PROPN cana-3316 110 49 the the DET cana-3316 110 50 pendent pendent NOUN cana-3316 110 51 vertices vertice VERB cana-3316 110 52 𝑒𝑖 𝑒𝑖 ADP cana-3316 110 53 for for ADP cana-3316 110 54 1 1 NUM cana-3316 110 55 ≀ ≀ NUM cana-3316 110 56 𝑖 𝑖 SYM cana-3316 110 57 ≀ ≀ NOUN cana-3316 110 58 𝑛 𝑛 PRON cana-3316 110 59 where where SCONJ cana-3316 110 60 𝑛 𝑛 PROPN cana-3316 110 61 β‰₯ β‰₯ NOUN cana-3316 110 62 4 4 NUM cana-3316 110 63 power power NOUN cana-3316 110 64 dominates dominate VERB cana-3316 110 65 𝑁[𝑒𝑖 𝑁[𝑒𝑖 NOUN cana-3316 110 66 ] ] PUNCT cana-3316 110 67 for for ADP cana-3316 110 68 1 1 NUM cana-3316 110 69 ≀ ≀ NUM cana-3316 110 70 𝑖 𝑖 SYM cana-3316 110 71 ≀ ≀ NOUN cana-3316 110 72 𝑛 𝑛 PRON cana-3316 110 73 where where SCONJ cana-3316 110 74 𝑛 𝑛 PROPN cana-3316 110 75 β‰₯ β‰₯ NOUN cana-3316 110 76 4 4 NUM cana-3316 110 77 . . PUNCT cana-3316 111 1 moreover moreover ADV cana-3316 111 2 , , PUNCT cana-3316 111 3 the the DET cana-3316 111 4 vertices vertex NOUN cana-3316 111 5 on on ADP cana-3316 111 6 the the DET cana-3316 111 7 cycle cycle NOUN cana-3316 111 8 π‘Žπ‘– π‘Žπ‘– ADP cana-3316 111 9 , , PUNCT cana-3316 111 10 1 1 NUM cana-3316 111 11 ≀ ≀ NUM cana-3316 111 12 𝑖 𝑖 SYM cana-3316 111 13 ≀ ≀ NOUN cana-3316 111 14 𝑛 𝑛 DET cana-3316 111 15 power power NOUN cana-3316 111 16 dominates dominate VERB cana-3316 111 17 𝑁[π‘Žπ‘– 𝑁[π‘Žπ‘– ADV cana-3316 111 18 ] ] X cana-3316 111 19 , , PUNCT cana-3316 111 20 1 1 NUM cana-3316 111 21 ≀ ≀ NUM cana-3316 111 22 𝑖 𝑖 SYM cana-3316 111 23 ≀ ≀ NOUN cana-3316 111 24 𝑛 𝑛 PRON cana-3316 111 25 where where SCONJ cana-3316 111 26 𝑛 𝑛 PROPN cana-3316 111 27 β‰₯ β‰₯ NOUN cana-3316 111 28 4 4 NUM cana-3316 111 29 . . PUNCT cana-3316 112 1 so so ADV cana-3316 112 2 , , PUNCT cana-3316 112 3 assign assign VERB cana-3316 112 4 color color NOUN cana-3316 112 5 𝑖 𝑖 X cana-3316 112 6 to to ADP cana-3316 112 7 { { PUNCT cana-3316 112 8 π‘Žπ‘– π‘Žπ‘– PROPN cana-3316 112 9 } } PUNCT cana-3316 112 10 , , PUNCT cana-3316 112 11 1 1 NUM cana-3316 112 12 ≀ ≀ NUM cana-3316 112 13 𝑖 𝑖 SYM cana-3316 112 14 ≀ ≀ NOUN cana-3316 112 15 𝑛 𝑛 PRON cana-3316 112 16 respectively respectively ADV cana-3316 112 17 . . PUNCT cana-3316 113 1 assign assign VERB cana-3316 113 2 ⌈ ⌈ PROPN cana-3316 113 3 𝑛+1 𝑛+1 ADP cana-3316 113 4 2 2 NUM cana-3316 113 5 βŒ‰ βŒ‰ VERB cana-3316 113 6 colors color NOUN cana-3316 113 7 to to ADP cana-3316 113 8 the the DET cana-3316 113 9 leftover leftover NOUN cana-3316 113 10 𝑛 𝑛 PROPN cana-3316 113 11 + + NOUN cana-3316 113 12 1 1 NUM cana-3316 113 13 vertices vertex NOUN cana-3316 113 14 equitably equitably ADV cana-3316 113 15 . . PUNCT cana-3316 114 1 the the DET cana-3316 114 2 central central ADJ cana-3316 114 3 vertex vertex NOUN cana-3316 114 4 π‘Žπ‘›+1 π‘Žπ‘›+1 NOUN cana-3316 114 5 power power NOUN cana-3316 114 6 dominates dominate VERB cana-3316 114 7 all all DET cana-3316 114 8 the the DET cana-3316 114 9 color color NOUN cana-3316 114 10 classes class NOUN cana-3316 114 11 . . PUNCT cana-3316 115 1 vertex vertex NOUN cana-3316 115 2 𝑒𝑖 𝑒𝑖 NOUN cana-3316 115 3 , , PUNCT cana-3316 115 4 1 1 NUM cana-3316 115 5 ≀ ≀ NUM cana-3316 115 6 𝑖 𝑖 SYM cana-3316 115 7 ≀ ≀ NUM cana-3316 115 8 𝑛 𝑛 PRON cana-3316 115 9 and and CCONJ cana-3316 115 10 π‘Žπ‘– π‘Žπ‘– CCONJ cana-3316 115 11 , , PUNCT cana-3316 115 12 1 1 NUM cana-3316 115 13 ≀ ≀ NUM cana-3316 115 14 𝑖 𝑖 SYM cana-3316 115 15 ≀ ≀ NOUN cana-3316 115 16 𝑛 𝑛 DET cana-3316 115 17 power power NOUN cana-3316 115 18 dominates dominate VERB cana-3316 115 19 the the DET cana-3316 115 20 color color NOUN cana-3316 115 21 class class NOUN cana-3316 115 22 𝐢𝑖 𝐢𝑖 PROPN cana-3316 115 23 , , PUNCT cana-3316 115 24 1 1 NUM cana-3316 115 25 ≀ ≀ NUM cana-3316 115 26 𝑖 𝑖 SYM cana-3316 115 27 ≀ ≀ NOUN cana-3316 115 28 𝑛 𝑛 ADP cana-3316 115 29 respectively respectively ADV cana-3316 115 30 and and CCONJ cana-3316 115 31 also also ADV cana-3316 115 32 it it PRON cana-3316 115 33 holds hold VERB cana-3316 115 34 the the DET cana-3316 115 35 inequality inequality NOUN cana-3316 115 36 ||𝐢𝑖| ||𝐢𝑖| VERB cana-3316 115 37 βˆ’ βˆ’ PROPN cana-3316 115 38 |𝐢𝑗|| |𝐢𝑗|| SYM cana-3316 115 39 ≀ ≀ NUM cana-3316 115 40 1 1 NUM cana-3316 115 41 , , PUNCT cana-3316 115 42 βˆ€ βˆ€ NOUN cana-3316 115 43 𝑖 𝑖 NOUN cana-3316 115 44 and and CCONJ cana-3316 115 45 𝑗. 𝑗. NOUN cana-3316 115 46 hence hence ADV cana-3316 115 47 πœ’π‘π‘‘π‘’(𝐻𝑛 πœ’π‘π‘‘π‘’(𝐻𝑛 PROPN cana-3316 115 48 ) ) PUNCT cana-3316 115 49 = = SYM cana-3316 115 50 𝑛 𝑛 PROPN cana-3316 115 51 + + SYM cana-3316 115 52 ⌈ ⌈ NOUN cana-3316 115 53 𝑛+1 𝑛+1 ADP cana-3316 115 54 2 2 NUM cana-3316 115 55 βŒ‰ βŒ‰ X cana-3316 115 56 5 5 NUM cana-3316 115 57 . . PUNCT cana-3316 115 58 conclusion conclusion VERB cana-3316 115 59 the the DET cana-3316 115 60 inequality inequality NOUN cana-3316 115 61 πœ’(𝐺 πœ’(𝐺 NOUN cana-3316 115 62 ) ) PUNCT cana-3316 115 63 ≀ ≀ NOUN cana-3316 115 64 πœ’π‘π‘‘(𝐺 πœ’π‘π‘‘(𝐺 NUM cana-3316 115 65 ) ) PUNCT cana-3316 115 66 ≀ ≀ NOUN cana-3316 115 67 πœ’π‘’(𝐺 πœ’π‘’(𝐺 NUM cana-3316 115 68 ) ) PUNCT cana-3316 115 69 ≀ ≀ NOUN cana-3316 115 70 πœ’π‘π‘‘π‘’(𝐺 πœ’π‘π‘‘π‘’(𝐺 X cana-3316 115 71 ) ) PUNCT cana-3316 115 72 provides provide VERB cana-3316 115 73 a a DET cana-3316 115 74 concise concise ADJ cana-3316 115 75 framework framework NOUN cana-3316 115 76 for for ADP cana-3316 115 77 understanding understand VERB cana-3316 115 78 the the DET cana-3316 115 79 precise precise ADJ cana-3316 115 80 values value NOUN cana-3316 115 81 of of ADP cana-3316 115 82 power power NOUN cana-3316 115 83 dominator dominator NOUN cana-3316 115 84 equitable equitable ADJ cana-3316 115 85 chromatic chromatic ADJ cana-3316 115 86 numbers number NOUN cana-3316 115 87 across across ADP cana-3316 115 88 various various ADJ cana-3316 115 89 standard standard ADJ cana-3316 115 90 graphs graph NOUN cana-3316 115 91 , , PUNCT cana-3316 115 92 such such ADJ cana-3316 115 93 as as ADP cana-3316 115 94 𝑃𝑛 𝑃𝑛 PROPN cana-3316 115 95 , , PUNCT cana-3316 115 96 𝐢𝑛 𝐢𝑛 PROPN cana-3316 115 97 , , PUNCT cana-3316 115 98 𝐾𝑛 𝐾𝑛 PROPN cana-3316 115 99 , , PUNCT cana-3316 115 100 πΎπ‘š,𝑛 πΎπ‘š,𝑛 PROPN cana-3316 115 101 , , PUNCT cana-3316 115 102 π‘Šπ‘› π‘Šπ‘› PROPN cana-3316 115 103 and and CCONJ cana-3316 115 104 𝐻𝑛. 𝐻𝑛. VERB cana-3316 115 105 the the DET cana-3316 115 106 graph graph NOUN cana-3316 115 107 𝐺 𝐺 PROPN cana-3316 115 108 , , PUNCT cana-3316 115 109 exhibiting exhibit VERB cana-3316 115 110 πœ’(𝐺 πœ’(𝐺 NOUN cana-3316 115 111 ) ) PUNCT cana-3316 115 112 = = SYM cana-3316 115 113 πœ’π‘π‘‘(𝐺 πœ’π‘π‘‘(𝐺 NUM cana-3316 115 114 ) ) PUNCT cana-3316 115 115 = = SYM cana-3316 115 116 πœ’π‘’(𝐺 πœ’π‘’(𝐺 NUM cana-3316 115 117 ) ) PUNCT cana-3316 115 118 = = SYM cana-3316 116 1 πœ’π‘π‘‘π‘’(𝐺 πœ’π‘π‘‘π‘’(𝐺 X cana-3316 116 2 ) ) PUNCT cana-3316 116 3 encompasses encompass VERB cana-3316 116 4 𝑃𝑛 𝑃𝑛 NOUN cana-3316 116 5 , , PUNCT cana-3316 116 6 𝐢𝑛 𝐢𝑛 PROPN cana-3316 116 7 and and CCONJ cana-3316 116 8 𝐾𝑛 𝐾𝑛 PROPN cana-3316 116 9 whereas whereas SCONJ cana-3316 116 10 for for ADP cana-3316 116 11 πΎπ‘š,𝑛 πΎπ‘š,𝑛 PROPN cana-3316 116 12 , , PUNCT cana-3316 116 13 all all DET cana-3316 116 14 these these DET cana-3316 116 15 parameters parameter NOUN cana-3316 116 16 are be AUX cana-3316 116 17 equal equal ADJ cana-3316 116 18 when when SCONJ cana-3316 116 19 ( ( PUNCT cana-3316 116 20 π‘š π‘š PROPN cana-3316 116 21 βˆ’ βˆ’ PROPN cana-3316 116 22 𝑛 𝑛 NOUN cana-3316 116 23 ) ) PUNCT cana-3316 116 24 < < X cana-3316 117 1 2 2 X cana-3316 117 2 . . X cana-3316 117 3 for for ADP cana-3316 117 4 wheel wheel NOUN cana-3316 117 5 graph graph NOUN cana-3316 117 6 π‘Šπ‘› π‘Šπ‘› PROPN cana-3316 117 7 , , PUNCT cana-3316 117 8 πœ’π‘’(π‘Šπ‘› πœ’π‘’(π‘Šπ‘› PROPN cana-3316 117 9 ) ) PUNCT cana-3316 117 10 = = SYM cana-3316 117 11 πœ’π‘π‘‘π‘’(π‘Šπ‘› πœ’π‘π‘‘π‘’(π‘Šπ‘› PROPN cana-3316 117 12 ) ) PUNCT cana-3316 117 13 . . PUNCT cana-3316 118 1 for for ADP cana-3316 118 2 helm helm NOUN cana-3316 118 3 graph graph NOUN cana-3316 118 4 𝐻𝑛 𝐻𝑛 ADJ cana-3316 118 5 , , PUNCT cana-3316 118 6 πœ’(𝐻𝑛 πœ’(𝐻𝑛 NUM cana-3316 118 7 ) ) PUNCT cana-3316 118 8 = = SYM cana-3316 118 9 πœ’π‘’(𝐻𝑛 πœ’π‘’(𝐻𝑛 NOUN cana-3316 118 10 ) ) PUNCT cana-3316 118 11 and and CCONJ cana-3316 118 12 πœ’π‘π‘‘π‘’(𝐻𝑛 πœ’π‘π‘‘π‘’(𝐻𝑛 PROPN cana-3316 118 13 ) ) PUNCT cana-3316 118 14 does do AUX cana-3316 118 15 not not PART cana-3316 118 16 coincide coincide VERB cana-3316 118 17 with with ADP cana-3316 118 18 other other ADJ cana-3316 118 19 three three NUM cana-3316 118 20 parameters parameter NOUN cana-3316 118 21 . . PUNCT cana-3316 119 1 determining determine VERB cana-3316 119 2 the the DET cana-3316 119 3 power power NOUN cana-3316 119 4 dominator dominator NOUN cana-3316 119 5 equitable equitable ADJ cana-3316 119 6 chromatic chromatic ADJ cana-3316 119 7 numbers number NOUN cana-3316 119 8 for for ADP cana-3316 119 9 diverse diverse ADJ cana-3316 119 10 graphs graph NOUN cana-3316 119 11 remains remain VERB cana-3316 119 12 a a DET cana-3316 119 13 task task NOUN cana-3316 119 14 reserved reserve VERB cana-3316 119 15 for for ADP cana-3316 119 16 future future ADJ cana-3316 119 17 investigation investigation NOUN cana-3316 119 18 . . PUNCT cana-3316 120 1 refrences refrence VERB cana-3316 121 1 [ [ X cana-3316 121 2 1 1 X cana-3316 121 3 ] ] X cana-3316 121 4 i i PRON cana-3316 121 5 chandramani chandramani PROPN cana-3316 121 6 , , PUNCT cana-3316 121 7 as as ADP cana-3316 121 8 prasanna prasanna PROPN cana-3316 121 9 venkatesan venkatesan NOUN cana-3316 121 10 , , PUNCT cana-3316 121 11 and and CCONJ cana-3316 121 12 sastha sastha PROPN cana-3316 121 13 sriram sriram PROPN cana-3316 121 14 . . PUNCT cana-3316 122 1 power power NOUN cana-3316 122 2 dominator dominator NOUN cana-3316 122 3 chromatic chromatic ADJ cana-3316 122 4 numbers number NOUN cana-3316 122 5 of of ADP cana-3316 122 6 jahangir jahangir PROPN cana-3316 122 7 and and CCONJ cana-3316 122 8 associated associated ADJ cana-3316 122 9 graph graph NOUN cana-3316 122 10 . . PUNCT cana-3316 122 11 2022 2022 NUM cana-3316 122 12 . . PUNCT cana-3316 123 1 communications communication NOUN cana-3316 123 2 on on ADP cana-3316 123 3 applied apply VERB cana-3316 123 4 nonlinear nonlinear ADJ cana-3316 123 5 analysis analysis NOUN cana-3316 123 6 issn issn NOUN cana-3316 123 7 : : PUNCT cana-3316 123 8 1074 1074 NUM cana-3316 123 9 - - PUNCT cana-3316 123 10 133x 133x NUM cana-3316 123 11 vol vol NOUN cana-3316 123 12 32 32 NUM cana-3316 123 13 no no NOUN cana-3316 123 14 . . PUNCT cana-3316 124 1 6s 6s NUM cana-3316 124 2 ( ( PUNCT cana-3316 124 3 2025 2025 NUM cana-3316 124 4 ) ) PUNCT cana-3316 124 5 534 534 NUM cana-3316 124 6 https://internationalpubls.com https://internationalpubls.com X cana-3316 125 1 [ [ X cana-3316 125 2 2 2 NUM cana-3316 125 3 ] ] PUNCT cana-3316 125 4 mustapha mustapha PROPN cana-3316 125 5 chellali chellali PROPN cana-3316 125 6 and and CCONJ cana-3316 125 7 frederic frederic PROPN cana-3316 125 8 maffray maffray PROPN cana-3316 125 9 . . PUNCT cana-3316 126 1 dominator dominator NOUN cana-3316 126 2 colorings coloring NOUN cana-3316 126 3 in in ADP cana-3316 126 4 some some DET cana-3316 126 5 classes class NOUN cana-3316 126 6 of of ADP cana-3316 126 7 graphs graph NOUN cana-3316 126 8 . . PUNCT cana-3316 127 1 graphs graph NOUN cana-3316 127 2 and and CCONJ cana-3316 127 3 combinatorics combinatoric NOUN cana-3316 127 4 , , PUNCT cana-3316 127 5 28:97–107 28:97–107 NUM cana-3316 127 6 , , PUNCT cana-3316 127 7 2012 2012 NUM cana-3316 127 8 . . PUNCT cana-3316 128 1 [ [ X cana-3316 128 2 3 3 X cana-3316 128 3 ] ] X cana-3316 128 4 michael michael PROPN cana-3316 128 5 dorfling dorfling PROPN cana-3316 128 6 and and CCONJ cana-3316 128 7 michael michael PROPN cana-3316 128 8 a a DET cana-3316 128 9 henning henning PROPN cana-3316 128 10 . . PUNCT cana-3316 129 1 a a DET cana-3316 129 2 note note NOUN cana-3316 129 3 on on ADP cana-3316 129 4 power power NOUN cana-3316 129 5 domination domination NOUN cana-3316 129 6 in in ADP cana-3316 129 7 grid grid NOUN cana-3316 129 8 graphs graph NOUN cana-3316 129 9 . . PUNCT cana-3316 130 1 discrete discrete ADJ cana-3316 130 2 applied apply VERB cana-3316 130 3 mathematics mathematic NOUN cana-3316 130 4 , , PUNCT cana-3316 130 5 154(6):1023–1027 154(6):1023–1027 NUM cana-3316 130 6 , , PUNCT cana-3316 130 7 2006 2006 NUM cana-3316 130 8 . . PUNCT cana-3316 131 1 [ [ X cana-3316 131 2 4 4 NUM cana-3316 131 3 ] ] X cana-3316 131 4 hanna hanna X cana-3316 131 5 furmaΒ΄nczyk furmaΒ΄nczyk NOUN cana-3316 131 6 . . PUNCT cana-3316 131 7 equitable equitable PROPN cana-3316 131 8 coloring coloring NOUN cana-3316 131 9 of of ADP cana-3316 131 10 graph graph NOUN cana-3316 131 11 products product NOUN cana-3316 131 12 . . PUNCT cana-3316 132 1 opuscula opuscula PROPN cana-3316 132 2 mathematica mathematica PROPN cana-3316 132 3 , , PUNCT cana-3316 132 4 26(1):31–44 26(1):31–44 NUM cana-3316 132 5 , , PUNCT cana-3316 132 6 2006 2006 NUM cana-3316 132 7 . . PUNCT cana-3316 133 1 [ [ X cana-3316 133 2 5 5 NUM cana-3316 133 3 ] ] PUNCT cana-3316 133 4 ralucca ralucca NOUN cana-3316 133 5 gera gera NOUN cana-3316 133 6 . . PUNCT cana-3316 134 1 on on ADP cana-3316 134 2 the the DET cana-3316 134 3 dominator dominator NOUN cana-3316 134 4 colorings coloring NOUN cana-3316 134 5 in in ADP cana-3316 134 6 bipartite bipartite NOUN cana-3316 134 7 graphs graph NOUN cana-3316 134 8 . . PUNCT cana-3316 135 1 in in ADP cana-3316 135 2 fourth fourth ADJ cana-3316 135 3 international international ADJ cana-3316 135 4 conference conference NOUN cana-3316 135 5 on on ADP cana-3316 135 6 information information NOUN cana-3316 135 7 technology technology NOUN cana-3316 135 8 ( ( PUNCT cana-3316 135 9 itng’07 itng’07 PROPN cana-3316 135 10 ) ) PUNCT cana-3316 135 11 , , PUNCT cana-3316 135 12 pages page NOUN cana-3316 135 13 947–952 947–952 NUM cana-3316 135 14 . . PUNCT cana-3316 136 1 ieee ieee NOUN cana-3316 136 2 , , PUNCT cana-3316 136 3 2007 2007 NUM cana-3316 136 4 . . PUNCT cana-3316 137 1 [ [ X cana-3316 137 2 6 6 NUM cana-3316 137 3 ] ] X cana-3316 137 4 frank frank PROPN cana-3316 137 5 harary harary PROPN cana-3316 137 6 . . PUNCT cana-3316 138 1 a a DET cana-3316 138 2 seminar seminar NOUN cana-3316 138 3 on on ADP cana-3316 138 4 graph graph NOUN cana-3316 138 5 theory theory NOUN cana-3316 138 6 . . PUNCT cana-3316 139 1 courier courier PROPN cana-3316 139 2 dover dover PROPN cana-3316 139 3 publications publication NOUN cana-3316 139 4 , , PUNCT cana-3316 139 5 2015 2015 NUM cana-3316 139 6 . . PUNCT cana-3316 140 1 [ [ X cana-3316 140 2 7 7 X cana-3316 140 3 ] ] X cana-3316 140 4 teresa teresa PROPN cana-3316 140 5 w w PROPN cana-3316 140 6 haynes haynes PROPN cana-3316 140 7 , , PUNCT cana-3316 140 8 sandra sandra PROPN cana-3316 140 9 m m PROPN cana-3316 140 10 hedetniemi hedetniemi ADV cana-3316 140 11 , , PUNCT cana-3316 140 12 stephen stephen PROPN cana-3316 140 13 t t PROPN cana-3316 140 14 hedetniemi hedetniemi ADV cana-3316 140 15 , , PUNCT cana-3316 140 16 and and CCONJ cana-3316 140 17 michael michael PROPN cana-3316 140 18 a a DET cana-3316 140 19 henning henning PROPN cana-3316 140 20 . . PUNCT cana-3316 141 1 domination domination NOUN cana-3316 141 2 in in ADP cana-3316 141 3 graphs graph NOUN cana-3316 141 4 applied apply VERB cana-3316 141 5 to to ADP cana-3316 141 6 electric electric ADJ cana-3316 141 7 power power NOUN cana-3316 141 8 networks network NOUN cana-3316 141 9 . . PUNCT cana-3316 142 1 siam siam PROPN cana-3316 142 2 journal journal PROPN cana-3316 142 3 on on ADP cana-3316 142 4 discrete discrete ADJ cana-3316 142 5 mathematics mathematic NOUN cana-3316 142 6 , , PUNCT cana-3316 142 7 15(4):519–529 15(4):519–529 NUM cana-3316 142 8 , , PUNCT cana-3316 142 9 2002 2002 NUM cana-3316 142 10 . . PUNCT cana-3316 143 1 [ [ X cana-3316 143 2 8 8 NUM cana-3316 143 3 ] ] X cana-3316 143 4 teresa teresa PROPN cana-3316 143 5 w w PROPN cana-3316 143 6 haynes haynes PROPN cana-3316 143 7 , , PUNCT cana-3316 143 8 stephen stephen PROPN cana-3316 143 9 t t PROPN cana-3316 143 10 hedetniemi hedetniemi ADV cana-3316 143 11 , , PUNCT cana-3316 143 12 and and CCONJ cana-3316 143 13 michael michael PROPN cana-3316 143 14 a a DET cana-3316 143 15 henning henning PROPN cana-3316 143 16 . . PUNCT cana-3316 144 1 domination domination NOUN cana-3316 144 2 in in ADP cana-3316 144 3 graphs graph NOUN cana-3316 144 4 : : PUNCT cana-3316 144 5 core core NOUN cana-3316 144 6 concepts concept NOUN cana-3316 144 7 . . PUNCT cana-3316 144 8 springer springer NOUN cana-3316 144 9 , , PUNCT cana-3316 144 10 2023 2023 NUM cana-3316 144 11 . . PUNCT cana-3316 145 1 [ [ X cana-3316 145 2 9 9 NUM cana-3316 145 3 ] ] X cana-3316 145 4 khee khee PROPN cana-3316 145 5 meng meng PROPN cana-3316 145 6 koh koh PROPN cana-3316 145 7 and and CCONJ cana-3316 145 8 kian kian PROPN cana-3316 145 9 wee wee PROPN cana-3316 145 10 soh soh PROPN cana-3316 145 11 . . PUNCT cana-3316 146 1 on on ADP cana-3316 146 2 the the DET cana-3316 146 3 power power NOUN cana-3316 146 4 domination domination NOUN cana-3316 146 5 number number NOUN cana-3316 146 6 of of ADP cana-3316 146 7 the the DET cana-3316 146 8 cartesian cartesian ADJ cana-3316 146 9 product product NOUN cana-3316 146 10 of of ADP cana-3316 146 11 graphs graph NOUN cana-3316 146 12 . . PUNCT cana-3316 147 1 akce akce PROPN cana-3316 147 2 international international PROPN cana-3316 147 3 journal journal NOUN cana-3316 147 4 of of ADP cana-3316 147 5 graphs graph NOUN cana-3316 147 6 and and CCONJ cana-3316 147 7 combinatorics combinatoric NOUN cana-3316 147 8 , , PUNCT cana-3316 147 9 16(3):253–257 16(3):253–257 PROPN cana-3316 147 10 , , PUNCT cana-3316 147 11 2019 2019 NUM cana-3316 147 12 . . PUNCT cana-3316 148 1 [ [ X cana-3316 148 2 10 10 NUM cana-3316 148 3 ] ] X cana-3316 148 4 a a DET cana-3316 148 5 uma uma PROPN cana-3316 148 6 maheswari maheswari PROPN cana-3316 148 7 and and CCONJ cana-3316 148 8 b b NOUN cana-3316 148 9 samuvel samuvel NOUN cana-3316 148 10 . . PUNCT cana-3316 149 1 power power NOUN cana-3316 149 2 dominator dominator NOUN cana-3316 149 3 chromatic chromatic ADJ cana-3316 149 4 number number NOUN cana-3316 149 5 for for ADP cana-3316 149 6 some some DET cana-3316 149 7 special special ADJ cana-3316 149 8 graphs graph NOUN cana-3316 149 9 . . PUNCT cana-3316 150 1 international international ADJ cana-3316 150 2 journal journal NOUN cana-3316 150 3 of of ADP cana-3316 150 4 innovative innovative ADJ cana-3316 150 5 technology technology NOUN cana-3316 150 6 and and CCONJ cana-3316 150 7 exploring explore VERB cana-3316 150 8 engineering engineering NOUN cana-3316 150 9 , , PUNCT cana-3316 150 10 8(12):3957–3960 8(12):3957–3960 NUM cana-3316 150 11 , , PUNCT cana-3316 150 12 2019 2019 NUM cana-3316 150 13 . . PUNCT cana-3316 151 1 [ [ X cana-3316 151 2 11 11 NUM cana-3316 151 3 ] ] X cana-3316 151 4 walter walter PROPN cana-3316 151 5 meyer meyer PROPN cana-3316 151 6 . . PROPN cana-3316 151 7 equitable equitable PROPN cana-3316 151 8 coloring color VERB cana-3316 151 9 . . PUNCT cana-3316 152 1 the the DET cana-3316 152 2 american american PROPN cana-3316 152 3 mathematical mathematical PROPN cana-3316 152 4 monthly monthly ADV cana-3316 152 5 , , PUNCT cana-3316 152 6 80(8):920–922 80(8):920–922 NUM cana-3316 152 7 , , PUNCT cana-3316 152 8 1973 1973 NUM cana-3316 152 9 . . PUNCT cana-3316 153 1 [ [ X cana-3316 153 2 12 12 NUM cana-3316 153 3 ] ] X cana-3316 153 4 minal minal ADJ cana-3316 153 5 shukla shukla NOUN cana-3316 153 6 and and CCONJ cana-3316 153 7 foram foram PROPN cana-3316 153 8 chandarana chandarana PROPN cana-3316 153 9 . . PUNCT cana-3316 154 1 dominator dominator NOUN cana-3316 154 2 coloring coloring NOUN cana-3316 154 3 of of ADP cana-3316 154 4 total total ADJ cana-3316 154 5 graph graph NOUN cana-3316 154 6 of of ADP cana-3316 154 7 path path NOUN cana-3316 154 8 and and CCONJ cana-3316 154 9 cycle cycle NOUN cana-3316 154 10 . . PUNCT cana-3316 155 1 mathematical mathematical ADJ cana-3316 155 2 models model NOUN cana-3316 155 3 in in ADP cana-3316 155 4 engineering engineering NOUN cana-3316 155 5 , , PUNCT cana-3316 155 6 9(2):72–80 9(2):72–80 NUM cana-3316 155 7 , , PUNCT cana-3316 155 8 2023 2023 NUM cana-3316 155 9 . . PUNCT cana-3316 156 1 [ [ X cana-3316 156 2 13 13 NUM cana-3316 156 3 ] ] SYM cana-3316 156 4 min min PROPN cana-3316 156 5 zhao zhao PROPN cana-3316 156 6 , , PUNCT cana-3316 156 7 liying liye VERB cana-3316 156 8 kang kang PROPN cana-3316 156 9 , , PUNCT cana-3316 156 10 and and CCONJ cana-3316 156 11 gerard gerard PROPN cana-3316 156 12 j j PROPN cana-3316 156 13 chang chang PROPN cana-3316 156 14 . . PUNCT cana-3316 157 1 power power NOUN cana-3316 157 2 domination domination NOUN cana-3316 157 3 in in ADP cana-3316 157 4 graphs graph NOUN cana-3316 157 5 . . PUNCT cana-3316 158 1 discrete discrete ADJ cana-3316 158 2 mathematics mathematic NOUN cana-3316 158 3 , , PUNCT cana-3316 158 4 306(15):1812 306(15):1812 NUM cana-3316 158 5 – – PUNCT cana-3316 158 6 1816 1816 NUM cana-3316 158 7 , , PUNCT cana-3316 158 8 2006 2006 NUM cana-3316 158 9 . . PUNCT