id sid tid token lemma pos cana-1219 1 1 communications communication NOUN cana-1219 1 2 on on ADP cana-1219 1 3 applied apply VERB cana-1219 1 4 nonlinear nonlinear ADJ cana-1219 1 5 analysis analysis NOUN cana-1219 1 6 issn issn NOUN cana-1219 1 7 : : PUNCT cana-1219 1 8 1074 1074 NUM cana-1219 1 9 - - PUNCT cana-1219 1 10 133x 133x NUM cana-1219 1 11 vol vol NOUN cana-1219 1 12 31 31 NUM cana-1219 1 13 no no NOUN cana-1219 1 14 . . PUNCT cana-1219 2 1 6s 6s NUM cana-1219 2 2 ( ( PUNCT cana-1219 2 3 2024 2024 NUM cana-1219 2 4 ) ) PUNCT cana-1219 2 5 252 252 NUM cana-1219 2 6 https://internationalpubls.com https://internationalpubls.com X cana-1219 2 7 evaluating evaluate VERB cana-1219 2 8 star star NOUN cana-1219 2 9 vertex vertex NOUN cana-1219 2 10 cochromatic cochromatic ADJ cana-1219 2 11 number number NOUN cana-1219 2 12 in in ADP cana-1219 2 13 prism prism NOUN cana-1219 2 14 , , PUNCT cana-1219 2 15 sunlet sunlet NOUN cana-1219 2 16 and and CCONJ cana-1219 2 17 derived derive VERB cana-1219 2 18 graphs graph NOUN cana-1219 2 19 r. r. PROPN cana-1219 2 20 sabitha1 sabitha1 PROPN cana-1219 2 21 * * PROPN cana-1219 2 22 , , PUNCT cana-1219 2 23 v. v. ADP cana-1219 2 24 kowsalya2 kowsalya2 PROPN cana-1219 3 1 1research 1research NUM cana-1219 3 2 scholar scholar NOUN cana-1219 3 3 , , PUNCT cana-1219 3 4 2associate 2associate NUM cana-1219 3 5 professor professor NOUN cana-1219 3 6 , , PUNCT cana-1219 3 7 pg pg PROPN cana-1219 3 8 & & CCONJ cana-1219 3 9 research research PROPN cana-1219 3 10 department department PROPN cana-1219 3 11 of of ADP cana-1219 3 12 mathematics mathematic NOUN cana-1219 3 13 , , PUNCT cana-1219 3 14 sri sri PROPN cana-1219 3 15 ramakrishna ramakrishna PROPN cana-1219 3 16 college college PROPN cana-1219 3 17 of of ADP cana-1219 3 18 arts arts PROPN cana-1219 3 19 & & CCONJ cana-1219 3 20 science science PROPN cana-1219 3 21 ( ( PUNCT cana-1219 3 22 autonomous autonomous ADJ cana-1219 3 23 ) ) PUNCT cana-1219 3 24 , , PUNCT cana-1219 3 25 formerly formerly ADV cana-1219 3 26 snr snr PROPN cana-1219 3 27 sons sons PROPN cana-1219 3 28 college college PROPN cana-1219 3 29 , , PUNCT cana-1219 3 30 coimbatore coimbatore PROPN cana-1219 3 31 . . PUNCT cana-1219 4 1 e e X cana-1219 4 2 - - NOUN cana-1219 4 3 mail mail NOUN cana-1219 4 4 i i NOUN cana-1219 4 5 d d NOUN cana-1219 4 6 : : PUNCT cana-1219 4 7 1*sabitha.r@srcas.ac.in 1*sabitha.r@srcas.ac.in NUM cana-1219 4 8 , , PUNCT cana-1219 4 9 2kowsalya@srcas.ac.in 2kowsalya@srcas.ac.in NUM cana-1219 4 10 article article NOUN cana-1219 4 11 history history NOUN cana-1219 4 12 : : PUNCT cana-1219 4 13 received receive VERB cana-1219 4 14 : : PUNCT cana-1219 4 15 12 12 NUM cana-1219 4 16 - - SYM cana-1219 4 17 01 01 NUM cana-1219 4 18 - - PUNCT cana-1219 4 19 2025 2025 NUM cana-1219 4 20 revised revise VERB cana-1219 4 21 : : PUNCT cana-1219 4 22 15 15 NUM cana-1219 4 23 - - NUM cana-1219 4 24 02 02 NUM cana-1219 4 25 - - PUNCT cana-1219 4 26 2025 2025 NUM cana-1219 4 27 accepted accept VERB cana-1219 4 28 : : PUNCT cana-1219 4 29 01 01 NUM cana-1219 4 30 - - SYM cana-1219 4 31 03 03 NUM cana-1219 4 32 - - PUNCT cana-1219 4 33 2025 2025 NUM cana-1219 4 34 abstract abstract NOUN cana-1219 4 35 : : PUNCT cana-1219 4 36 in in ADP cana-1219 4 37 this this DET cana-1219 4 38 discussion discussion NOUN cana-1219 4 39 , , PUNCT cana-1219 4 40 the the DET cana-1219 4 41 star star NOUN cana-1219 4 42 cochromatic cochromatic ADJ cana-1219 4 43 number number NOUN cana-1219 4 44 π‘ž π‘ž PROPN cana-1219 4 45 is be AUX cana-1219 4 46 found find VERB cana-1219 4 47 for for ADP cana-1219 4 48 the the DET cana-1219 4 49 following follow VERB cana-1219 4 50 graphs graph NOUN cana-1219 4 51 : : PUNCT cana-1219 4 52 prism prism NOUN cana-1219 4 53 graph graph NOUN cana-1219 4 54 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š PROPN cana-1219 4 55 ] ] X cana-1219 4 56 , , PUNCT cana-1219 4 57 line line NOUN cana-1219 4 58 graph graph NOUN cana-1219 4 59 of of ADP cana-1219 4 60 prism prism NOUN cana-1219 4 61 graph graph NOUN cana-1219 4 62 π‘ž[𝐿(π‘Œπ‘š π‘ž[𝐿(π‘Œπ‘š PROPN cana-1219 4 63 ) ) PUNCT cana-1219 4 64 ] ] PUNCT cana-1219 4 65 , , PUNCT cana-1219 4 66 middle middle ADJ cana-1219 4 67 graph graph NOUN cana-1219 4 68 of of ADP cana-1219 4 69 prism prism NOUN cana-1219 4 70 graph graph NOUN cana-1219 4 71 π‘ž[𝑀(π‘Œπ‘š π‘ž[𝑀(π‘Œπ‘š PROPN cana-1219 4 72 ) ) PUNCT cana-1219 4 73 ] ] PUNCT cana-1219 4 74 , , PUNCT cana-1219 4 75 sunlet sunlet NOUN cana-1219 4 76 graph graph NOUN cana-1219 4 77 π‘ž[π‘†π‘š π‘ž[π‘†π‘š PROPN cana-1219 4 78 ] ] X cana-1219 4 79 , , PUNCT cana-1219 4 80 line line NOUN cana-1219 4 81 graph graph NOUN cana-1219 4 82 of of ADP cana-1219 4 83 sunlet sunlet NOUN cana-1219 4 84 graph graph NOUN cana-1219 4 85 π‘ž[𝐿(π‘†π‘š π‘ž[𝐿(π‘†π‘š PROPN cana-1219 4 86 ) ) PUNCT cana-1219 4 87 ] ] PUNCT cana-1219 4 88 , , PUNCT cana-1219 4 89 middle middle ADJ cana-1219 4 90 graph graph NOUN cana-1219 4 91 of of ADP cana-1219 4 92 sunlet sunlet NOUN cana-1219 4 93 graph graph NOUN cana-1219 4 94 π‘ž[𝑀(π‘†π‘š π‘ž[𝑀(π‘†π‘š PROPN cana-1219 4 95 ) ) PUNCT cana-1219 4 96 ] ] PUNCT cana-1219 4 97 . . PUNCT cana-1219 5 1 keywords keyword NOUN cana-1219 5 2 : : PUNCT cana-1219 5 3 prism prism NOUN cana-1219 5 4 graph graph NOUN cana-1219 5 5 , , PUNCT cana-1219 5 6 sunlet sunlet NOUN cana-1219 5 7 graph graph NOUN cana-1219 5 8 , , PUNCT cana-1219 5 9 line line NOUN cana-1219 5 10 graph graph NOUN cana-1219 5 11 , , PUNCT cana-1219 5 12 middle middle ADJ cana-1219 5 13 graph graph NOUN cana-1219 5 14 , , PUNCT cana-1219 5 15 star star NOUN cana-1219 5 16 cocoloring cocoloring NOUN cana-1219 5 17 , , PUNCT cana-1219 5 18 star star NOUN cana-1219 5 19 cochromatic cochromatic PROPN cana-1219 5 20 index index PROPN cana-1219 5 21 . . PUNCT cana-1219 6 1 ams am NOUN cana-1219 6 2 classification classification NOUN cana-1219 6 3 : : PUNCT cana-1219 6 4 05c07 05c07 NOUN cana-1219 6 5 , , PUNCT cana-1219 6 6 05c15 05c15 NOUN cana-1219 6 7 , , PUNCT cana-1219 6 8 05c69 05c69 NUM cana-1219 6 9 , , PUNCT cana-1219 6 10 05c76 05c76 PRON cana-1219 6 11 . . PUNCT cana-1219 7 1 1 1 X cana-1219 7 2 . . X cana-1219 7 3 introduction introduction NOUN cana-1219 7 4 this this DET cana-1219 7 5 study study NOUN cana-1219 7 6 examines examine VERB cana-1219 7 7 finite finite NOUN cana-1219 7 8 , , PUNCT cana-1219 7 9 simple simple ADJ cana-1219 7 10 and and CCONJ cana-1219 7 11 undirected undirected ADJ cana-1219 7 12 graphs graph NOUN cana-1219 7 13 . . PUNCT cana-1219 8 1 whitney whitney PROPN cana-1219 8 2 initially initially ADV cana-1219 8 3 presented present VERB cana-1219 8 4 the the DET cana-1219 8 5 concept concept NOUN cana-1219 8 6 of of ADP cana-1219 8 7 a a DET cana-1219 8 8 line line NOUN cana-1219 8 9 graph graph NOUN cana-1219 8 10 in in ADP cana-1219 8 11 1932 1932 NUM cana-1219 8 12 [ [ X cana-1219 8 13 1 1 NUM cana-1219 8 14 ] ] PUNCT cana-1219 8 15 . . PUNCT cana-1219 9 1 in in ADP cana-1219 9 2 1983 1983 NUM cana-1219 9 3 , , PUNCT cana-1219 9 4 maria maria PROPN cana-1219 9 5 chudnovsky chudnovsky NOUN cana-1219 10 1 [ [ X cana-1219 10 2 5 5 NUM cana-1219 10 3 ] ] PUNCT cana-1219 10 4 proposed propose VERB cana-1219 10 5 using use VERB cana-1219 10 6 line line NOUN cana-1219 10 7 graphs graph NOUN cana-1219 10 8 as as ADP cana-1219 10 9 a a DET cana-1219 10 10 very very ADV cana-1219 10 11 fundamental fundamental ADJ cana-1219 10 12 lesson lesson NOUN cana-1219 10 13 in in ADP cana-1219 10 14 graph graph NOUN cana-1219 10 15 theory theory NOUN cana-1219 10 16 . . PUNCT cana-1219 11 1 t. t. PROPN cana-1219 11 2 hamada hamada PROPN cana-1219 11 3 and and CCONJ cana-1219 11 4 i. i. PROPN cana-1219 11 5 yoshimura yoshimura PROPN cana-1219 11 6 were be AUX cana-1219 11 7 the the DET cana-1219 11 8 ones one NOUN cana-1219 11 9 who who PRON cana-1219 11 10 first first ADV cana-1219 11 11 suggested suggest VERB cana-1219 11 12 the the DET cana-1219 11 13 center center ADJ cana-1219 11 14 graph graph NOUN cana-1219 11 15 of of ADP cana-1219 11 16 the the DET cana-1219 11 17 graph graph NOUN cana-1219 11 18 [ [ X cana-1219 11 19 6 6 NUM cana-1219 11 20 ] ] PUNCT cana-1219 11 21 . . PUNCT cana-1219 12 1 vertex vertex NOUN cana-1219 12 2 partitioning partitioning NOUN cana-1219 12 3 is be AUX cana-1219 12 4 an an DET cana-1219 12 5 important important ADJ cana-1219 12 6 concept concept NOUN cana-1219 12 7 in in ADP cana-1219 12 8 graph graph NOUN cana-1219 12 9 theory theory NOUN cana-1219 12 10 . . PUNCT cana-1219 13 1 partitioning partition VERB cana-1219 13 2 and and CCONJ cana-1219 13 3 imposing impose VERB cana-1219 13 4 on on ADP cana-1219 13 5 vertex vertex NOUN cana-1219 13 6 sets set NOUN cana-1219 13 7 yielded yield VERB cana-1219 13 8 new new ADJ cana-1219 13 9 concepts concept NOUN cana-1219 13 10 and and CCONJ cana-1219 13 11 outcomes outcome NOUN cana-1219 13 12 . . PUNCT cana-1219 14 1 in in ADP cana-1219 14 2 1977 1977 NUM cana-1219 14 3 , , PUNCT cana-1219 14 4 lesinak lesinak NOUN cana-1219 14 5 and and CCONJ cana-1219 14 6 h. h. PROPN cana-1219 14 7 straight straight PROPN cana-1219 14 8 introduced introduce VERB cana-1219 14 9 the the DET cana-1219 14 10 concept concept NOUN cana-1219 14 11 cocoloring cocolore VERB cana-1219 14 12 and and CCONJ cana-1219 14 13 found find VERB cana-1219 14 14 some some DET cana-1219 14 15 basic basic ADJ cana-1219 14 16 results result NOUN cana-1219 14 17 [ [ X cana-1219 14 18 3 3 NUM cana-1219 14 19 ] ] PUNCT cana-1219 14 20 . . PUNCT cana-1219 15 1 kowsalya kowsalya PROPN cana-1219 15 2 . . PUNCT cana-1219 16 1 v v X cana-1219 16 2 , , PUNCT cana-1219 16 3 vernold vernold ADJ cana-1219 16 4 vivin vivin NOUN cana-1219 16 5 . . PUNCT cana-1219 17 1 j j PROPN cana-1219 17 2 and and CCONJ cana-1219 17 3 venkatachalam venkatachalam PROPN cana-1219 17 4 . . PUNCT cana-1219 18 1 m m VERB cana-1219 19 1 [ [ X cana-1219 19 2 2 2 NUM cana-1219 19 3 ] ] PUNCT cana-1219 19 4 found find VERB cana-1219 19 5 the the DET cana-1219 19 6 star star ADJ cana-1219 19 7 chromatic chromatic ADJ cana-1219 19 8 number number NOUN cana-1219 19 9 for for ADP cana-1219 19 10 sunlet sunlet NOUN cana-1219 19 11 graph graph NOUN cana-1219 19 12 families family NOUN cana-1219 19 13 and and CCONJ cana-1219 19 14 their their PRON cana-1219 19 15 line line NOUN cana-1219 19 16 , , PUNCT cana-1219 19 17 middle middle ADJ cana-1219 19 18 , , PUNCT cana-1219 19 19 central central ADJ cana-1219 19 20 and and CCONJ cana-1219 19 21 total total ADJ cana-1219 19 22 graphs graph NOUN cana-1219 19 23 . . PUNCT cana-1219 20 1 vernold vernold PROPN cana-1219 20 2 vivin vivin PROPN cana-1219 20 3 . . PUNCT cana-1219 21 1 j j PROPN cana-1219 21 2 , , PUNCT cana-1219 21 3 kowsalya kowsalya PROPN cana-1219 21 4 . . PUNCT cana-1219 22 1 v v NOUN cana-1219 22 2 , , PUNCT cana-1219 22 3 and and CCONJ cana-1219 22 4 vimal vimal PROPN cana-1219 22 5 kumar kumar PROPN cana-1219 22 6 . . PUNCT cana-1219 22 7 s s PART cana-1219 23 1 [ [ X cana-1219 23 2 7 7 NUM cana-1219 23 3 ] ] PUNCT cana-1219 23 4 found find VERB cana-1219 23 5 the the DET cana-1219 23 6 star star ADJ cana-1219 23 7 chromatic chromatic ADJ cana-1219 23 8 number number NOUN cana-1219 23 9 for for ADP cana-1219 23 10 prism prism NOUN cana-1219 23 11 graph graph NOUN cana-1219 23 12 families family NOUN cana-1219 23 13 and and CCONJ cana-1219 23 14 their their PRON cana-1219 23 15 derived derive VERB cana-1219 23 16 graphs graph NOUN cana-1219 23 17 . . PUNCT cana-1219 24 1 star star NOUN cana-1219 24 2 cocoloring cocolore VERB cana-1219 24 3 concept concept NOUN cana-1219 24 4 was be AUX cana-1219 24 5 introduced introduce VERB cana-1219 24 6 by by ADP cana-1219 24 7 m. m. NOUN cana-1219 24 8 poobalaranjani[4 poobalaranjani[4 PROPN cana-1219 24 9 ] ] PUNCT cana-1219 24 10 . . PUNCT cana-1219 25 1 a a DET cana-1219 25 2 clique clique NOUN cana-1219 25 3 is be AUX cana-1219 25 4 defined define VERB cana-1219 25 5 as as ADP cana-1219 25 6 a a DET cana-1219 25 7 subset subset NOUN cana-1219 25 8 w w NOUN cana-1219 25 9 of of ADP cana-1219 25 10 a a DET cana-1219 25 11 simple simple ADJ cana-1219 25 12 graph graph NOUN cana-1219 25 13 g g PROPN cana-1219 25 14 = = SYM cana-1219 25 15 ( ( PUNCT cana-1219 25 16 v v NOUN cana-1219 25 17 , , PUNCT cana-1219 25 18 e e NOUN cana-1219 25 19 ) ) PUNCT cana-1219 25 20 that that PRON cana-1219 25 21 produces produce VERB cana-1219 25 22 a a DET cana-1219 25 23 complete complete ADJ cana-1219 25 24 subgraph subgraph NOUN cana-1219 25 25 of of ADP cana-1219 25 26 g. g. PROPN cana-1219 25 27 if if SCONJ cana-1219 25 28 w w PROPN cana-1219 25 29 has have VERB cana-1219 25 30 cardinality cardinality PROPN cana-1219 25 31 k k NOUN cana-1219 25 32 , , PUNCT cana-1219 25 33 it it PRON cana-1219 25 34 is be AUX cana-1219 25 35 referred refer VERB cana-1219 25 36 to to ADP cana-1219 25 37 as as ADP cana-1219 25 38 k k NOUN cana-1219 25 39 - - NOUN cana-1219 25 40 clique clique NOUN cana-1219 25 41 . . PUNCT cana-1219 26 1 a a DET cana-1219 26 2 subset subset ADJ cana-1219 26 3 u u NOUN cana-1219 26 4 of of ADP cana-1219 26 5 v v NOUN cana-1219 26 6 is be AUX cana-1219 26 7 considered consider VERB cana-1219 26 8 independent independent ADJ cana-1219 26 9 if if SCONJ cana-1219 26 10 it it PRON cana-1219 26 11 creates create VERB cana-1219 26 12 an an DET cana-1219 26 13 empty empty ADJ cana-1219 26 14 subgraph subgraph NOUN cana-1219 26 15 of of ADP cana-1219 26 16 g. g. PROPN cana-1219 26 17 if if SCONJ cana-1219 26 18 u u NOUN cana-1219 26 19 has have VERB cana-1219 26 20 cardinality cardinality PROPN cana-1219 26 21 k k NOUN cana-1219 26 22 , , PUNCT cana-1219 26 23 it it PRON cana-1219 26 24 is be AUX cana-1219 26 25 referred refer VERB cana-1219 26 26 to to ADP cana-1219 26 27 k k ADJ cana-1219 26 28 - - ADJ cana-1219 26 29 independent independent ADJ cana-1219 26 30 set set NOUN cana-1219 26 31 . . PUNCT cana-1219 27 1 2 2 X cana-1219 27 2 . . X cana-1219 27 3 preliminaries preliminary NOUN cana-1219 27 4 2.1 2.1 NUM cana-1219 27 5 : : PUNCT cana-1219 27 6 prism prism NOUN cana-1219 27 7 graph graph NOUN cana-1219 27 8 : : PUNCT cana-1219 27 9 the the DET cana-1219 27 10 prism prism NOUN cana-1219 27 11 graph graph NOUN cana-1219 27 12 is be AUX cana-1219 27 13 a a DET cana-1219 27 14 planar planar ADJ cana-1219 27 15 , , PUNCT cana-1219 27 16 polyhedral polyhedral ADJ cana-1219 27 17 graph graph NOUN cana-1219 27 18 that that PRON cana-1219 27 19 resembles resemble VERB cana-1219 27 20 the the DET cana-1219 27 21 skeleton skeleton NOUN cana-1219 27 22 of of ADP cana-1219 27 23 an an DET cana-1219 27 24 mprism mprism NOUN cana-1219 27 25 . . PUNCT cana-1219 28 1 an an DET cana-1219 28 2 m m ADJ cana-1219 28 3 - - ADJ cana-1219 28 4 prism prism NOUN cana-1219 28 5 graph graph NOUN cana-1219 28 6 is be AUX cana-1219 28 7 the the DET cana-1219 28 8 same same ADJ cana-1219 28 9 as as ADP cana-1219 28 10 the the DET cana-1219 28 11 generalised generalised ADJ cana-1219 28 12 petersen petersen NOUN cana-1219 28 13 graph graph NOUN cana-1219 28 14 𝑃{𝑛,1 𝑃{𝑛,1 PROPN cana-1219 28 15 } } PUNCT cana-1219 28 16 with with ADP cana-1219 28 17 2 2 NUM cana-1219 28 18 m m NOUN cana-1219 28 19 vertices vertex NOUN cana-1219 28 20 and and CCONJ cana-1219 28 21 3 3 NUM cana-1219 28 22 m m NOUN cana-1219 28 23 edges edge NOUN cana-1219 28 24 . . PUNCT cana-1219 29 1 it it PRON cana-1219 29 2 is be AUX cana-1219 29 3 denoted denote VERB cana-1219 29 4 by by ADP cana-1219 29 5 π‘Œπ‘š π‘Œπ‘š PROPN cana-1219 29 6 2.2 2.2 NUM cana-1219 29 7 : : PUNCT cana-1219 29 8 sunlet sunlet NOUN cana-1219 29 9 graph graph NOUN cana-1219 29 10 : : PUNCT cana-1219 29 11 the the DET cana-1219 29 12 sunlet sunlet NOUN cana-1219 29 13 graph graph NOUN cana-1219 29 14 on on ADP cana-1219 29 15 2 2 NUM cana-1219 29 16 m m NOUN cana-1219 29 17 vertices vertex NOUN cana-1219 29 18 is be AUX cana-1219 29 19 obtained obtain VERB cana-1219 29 20 by by ADP cana-1219 29 21 attaching attach VERB cana-1219 29 22 n n DET cana-1219 29 23 pendant pendant ADJ cana-1219 29 24 edges edge NOUN cana-1219 29 25 to to ADP cana-1219 29 26 the the DET cana-1219 29 27 cycle cycle NOUN cana-1219 30 1 𝐢𝑛 𝐢𝑛 PROPN cana-1219 30 2 and and CCONJ cana-1219 30 3 it it PRON cana-1219 30 4 is be AUX cana-1219 30 5 denoted denote VERB cana-1219 30 6 by by ADP cana-1219 30 7 π‘†π‘š. π‘†π‘š. PROPN cana-1219 30 8 mailto:1*sabitha.r@srcas.ac.in mailto:1*sabitha.r@srcas.ac.in NOUN cana-1219 30 9 mailto:2kowsalya@srcas.ac.in mailto:2kowsalya@srcas.ac.in NOUN cana-1219 30 10 communications communication NOUN cana-1219 30 11 on on ADP cana-1219 30 12 applied apply VERB cana-1219 30 13 nonlinear nonlinear ADJ cana-1219 30 14 analysis analysis NOUN cana-1219 30 15 issn issn NOUN cana-1219 30 16 : : PUNCT cana-1219 30 17 1074 1074 NUM cana-1219 30 18 - - PUNCT cana-1219 30 19 133x 133x NUM cana-1219 30 20 vol vol NOUN cana-1219 30 21 31 31 NUM cana-1219 30 22 no no NOUN cana-1219 30 23 . . PUNCT cana-1219 31 1 6s 6s NUM cana-1219 31 2 ( ( PUNCT cana-1219 31 3 2024 2024 NUM cana-1219 31 4 ) ) PUNCT cana-1219 31 5 253 253 NUM cana-1219 31 6 https://internationalpubls.com https://internationalpubls.com X cana-1219 31 7 2.3 2.3 NUM cana-1219 31 8 : : PUNCT cana-1219 31 9 line line NOUN cana-1219 31 10 graph graph NOUN cana-1219 31 11 : : PUNCT cana-1219 31 12 line line NOUN cana-1219 31 13 graph graph NOUN cana-1219 31 14 l(g l(g PROPN cana-1219 31 15 ) ) PUNCT cana-1219 31 16 is be AUX cana-1219 31 17 β€’ β€’ X cana-1219 31 18 l(g l(g PROPN cana-1219 31 19 ) ) PUNCT cana-1219 31 20 is be AUX cana-1219 31 21 a a DET cana-1219 31 22 graph graph NOUN cana-1219 31 23 in in ADP cana-1219 31 24 which which PRON cana-1219 31 25 each each DET cana-1219 31 26 vertex vertex NOUN cana-1219 31 27 corresponds correspond VERB cana-1219 31 28 to to ADP cana-1219 31 29 an an DET cana-1219 31 30 edge edge NOUN cana-1219 31 31 of of ADP cana-1219 31 32 g. g. PROPN cana-1219 32 1 β€’ β€’ NUM cana-1219 32 2 two two NUM cana-1219 32 3 l(g l(g NOUN cana-1219 32 4 ) ) PUNCT cana-1219 32 5 vertices vertex NOUN cana-1219 32 6 are be AUX cana-1219 32 7 said say VERB cana-1219 32 8 to to PART cana-1219 32 9 be be AUX cana-1219 32 10 adjacent adjacent ADJ cana-1219 32 11 if if SCONJ cana-1219 32 12 their their PRON cana-1219 32 13 respective respective ADJ cana-1219 32 14 lines line NOUN cana-1219 32 15 share share VERB cana-1219 32 16 an an DET cana-1219 32 17 endpoint endpoint NOUN cana-1219 32 18 in in ADP cana-1219 32 19 g. g. PROPN cana-1219 32 20 2.4 2.4 NUM cana-1219 32 21 : : PUNCT cana-1219 32 22 middle middle ADJ cana-1219 32 23 graph graph NOUN cana-1219 32 24 : : PUNCT cana-1219 32 25 the the DET cana-1219 32 26 newly newly ADV cana-1219 32 27 added add VERB cana-1219 32 28 middle middle ADJ cana-1219 32 29 vertices vertex NOUN cana-1219 32 30 of of ADP cana-1219 32 31 g g PROPN cana-1219 32 32 's 's PART cana-1219 32 33 surrounding surround VERB cana-1219 32 34 edges edge NOUN cana-1219 32 35 are be AUX cana-1219 32 36 joined join VERB cana-1219 32 37 to to PART cana-1219 32 38 form form VERB cana-1219 32 39 the the DET cana-1219 32 40 middle middle ADJ cana-1219 32 41 graph graph NOUN cana-1219 32 42 m(g m(g PROPN cana-1219 32 43 ) ) PUNCT cana-1219 32 44 . . PUNCT cana-1219 33 1 it it PRON cana-1219 33 2 is be AUX cana-1219 33 3 made make VERB cana-1219 33 4 by by ADP cana-1219 33 5 accurately accurately ADV cana-1219 33 6 splitting split VERB cana-1219 33 7 each each DET cana-1219 33 8 edge edge NOUN cana-1219 33 9 of of ADP cana-1219 33 10 g g PROPN cana-1219 33 11 once once ADV cana-1219 33 12 . . PUNCT cana-1219 34 1 2.5 2.5 NUM cana-1219 34 2 : : PUNCT cana-1219 34 3 star star NOUN cana-1219 34 4 cocoloring cocoloring NOUN cana-1219 34 5 : : PUNCT cana-1219 34 6 let let VERB cana-1219 34 7 g g PRON cana-1219 34 8 be be AUX cana-1219 34 9 a a DET cana-1219 34 10 graph graph NOUN cana-1219 34 11 k k PROPN cana-1219 34 12 , , PUNCT cana-1219 34 13 l l NOUN cana-1219 34 14 , , PUNCT cana-1219 34 15 r r NOUN cana-1219 34 16 be be VERB cana-1219 34 17 non non ADJ cana-1219 34 18 - - ADJ cana-1219 34 19 negative negative ADJ cana-1219 34 20 integers integer NOUN cana-1219 34 21 then then ADV cana-1219 34 22 a a DET cana-1219 34 23 ( ( PUNCT cana-1219 34 24 k k X cana-1219 34 25 , , PUNCT cana-1219 34 26 l l NOUN cana-1219 34 27 , , PUNCT cana-1219 34 28 r r NOUN cana-1219 34 29 ) ) PUNCT cana-1219 34 30 s s AUX cana-1219 34 31 cocoloring cocoloring NOUN cana-1219 34 32 of of ADP cana-1219 34 33 g g PROPN cana-1219 34 34 is be AUX cana-1219 34 35 a a DET cana-1219 34 36 partition partition NOUN cana-1219 34 37 of of ADP cana-1219 34 38 the the DET cana-1219 34 39 vertex vertex NOUN cana-1219 34 40 set set NOUN cana-1219 34 41 of of ADP cana-1219 34 42 g g NOUN cana-1219 34 43 into into ADP cana-1219 34 44 sets set NOUN cana-1219 34 45 𝐼1 𝐼1 NOUN cana-1219 34 46 , , PUNCT cana-1219 34 47 𝐼2 𝐼2 NOUN cana-1219 34 48 , , PUNCT cana-1219 34 49 𝐼3 𝐼3 PROPN cana-1219 34 50 , , PUNCT cana-1219 34 51 … … PUNCT cana-1219 34 52 , , PUNCT cana-1219 34 53 πΌπ‘˜ πΌπ‘˜ PROPN cana-1219 34 54 , , PUNCT cana-1219 34 55 𝐢1 𝐢1 NUM cana-1219 34 56 , , PUNCT cana-1219 34 57 𝐢2 𝐢2 ADJ cana-1219 34 58 , , PUNCT cana-1219 34 59 𝐢3 𝐢3 NOUN cana-1219 34 60 , , PUNCT cana-1219 34 61 … … PUNCT cana-1219 34 62 , , PUNCT cana-1219 34 63 𝐢𝑙 𝐢𝑙 NOUN cana-1219 34 64 , , PUNCT cana-1219 34 65 𝑆1 𝑆1 PROPN cana-1219 34 66 , , PUNCT cana-1219 34 67 𝑆2 𝑆2 PROPN cana-1219 34 68 , , PUNCT cana-1219 34 69 𝑆3 𝑆3 PROPN cana-1219 34 70 , , PUNCT cana-1219 34 71 … … PUNCT cana-1219 34 72 , , PUNCT cana-1219 34 73 π‘†π‘Ÿ π‘†π‘Ÿ ADP cana-1219 34 74 such such ADJ cana-1219 34 75 that that SCONJ cana-1219 34 76 πΌπ‘˜ πΌπ‘˜ PROPN cana-1219 34 77 is be AUX cana-1219 34 78 an an DET cana-1219 34 79 independent independent ADJ cana-1219 34 80 set set NOUN cana-1219 34 81 each each DET cana-1219 34 82 𝐢𝑙 𝐢𝑙 PROPN cana-1219 34 83 is be AUX cana-1219 34 84 a a DET cana-1219 34 85 clique clique NOUN cana-1219 34 86 and and CCONJ cana-1219 34 87 each each DET cana-1219 34 88 π‘†π‘Ÿ π‘†π‘Ÿ PROPN cana-1219 34 89 is be AUX cana-1219 34 90 a a DET cana-1219 34 91 star star NOUN cana-1219 34 92 𝐾{1,𝑑 𝐾{1,𝑑 PROPN cana-1219 34 93 } } PUNCT cana-1219 34 94 where where SCONJ cana-1219 34 95 𝑑 𝑑 PROPN cana-1219 34 96 β‰₯ β‰₯ NOUN cana-1219 34 97 3 3 NUM cana-1219 34 98 . . NOUN cana-1219 34 99 2.6 2.6 NUM cana-1219 34 100 : : PUNCT cana-1219 34 101 star star NOUN cana-1219 34 102 cochromatic cochromatic ADJ cana-1219 34 103 number number NOUN cana-1219 34 104 : : PUNCT cana-1219 34 105 the the DET cana-1219 34 106 star star NOUN cana-1219 34 107 cochromatic cochromatic ADJ cana-1219 34 108 number number NOUN cana-1219 34 109 π‘ž π‘ž PROPN cana-1219 34 110 is be AUX cana-1219 34 111 defined define VERB cana-1219 34 112 as as ADP cana-1219 34 113 π‘§βˆ—(𝐺 π‘§βˆ—(𝐺 ADJ cana-1219 34 114 ) ) PUNCT cana-1219 34 115 = = SYM cana-1219 34 116 min{π‘ž min{π‘ž X cana-1219 34 117 : : PUNCT cana-1219 34 118 π‘‘β„Žπ‘’π‘Ÿπ‘’ π‘‘β„Žπ‘’π‘Ÿπ‘’ VERB cana-1219 34 119 𝑒π‘₯𝑖𝑠𝑑𝑠 𝑒π‘₯𝑖𝑠𝑑𝑠 PROPN cana-1219 34 120 π‘˜ π‘˜ PROPN cana-1219 34 121 , , PUNCT cana-1219 34 122 𝑙 𝑙 PRON cana-1219 34 123 , , PUNCT cana-1219 34 124 π‘Ÿ π‘Ÿ PRON cana-1219 34 125 π‘ π‘’π‘β„Ž π‘ π‘’π‘β„Ž VERB cana-1219 34 126 π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘Žπ‘‘ PROPN cana-1219 34 127 π‘˜ π‘˜ PROPN cana-1219 34 128 , , PUNCT cana-1219 34 129 𝑙 𝑙 PRON cana-1219 34 130 β‰₯ β‰₯ NOUN cana-1219 34 131 0 0 NUM cana-1219 34 132 , , PUNCT cana-1219 34 133 π‘Ÿ π‘Ÿ PRON cana-1219 34 134 β‰₯ β‰₯ NUM cana-1219 34 135 1 1 NUM cana-1219 34 136 ; ; PUNCT cana-1219 35 1 𝐺 𝐺 PROPN cana-1219 35 2 𝑖𝑠 𝑖𝑠 PROPN cana-1219 35 3 ( ( PUNCT cana-1219 35 4 π‘˜ π‘˜ PROPN cana-1219 35 5 , , PUNCT cana-1219 35 6 𝑙 𝑙 X cana-1219 35 7 , , PUNCT cana-1219 35 8 π‘Ÿ π‘Ÿ NOUN cana-1219 35 9 ) ) PUNCT cana-1219 35 10 βˆ’ βˆ’ NOUN cana-1219 36 1 𝑠 𝑠 INTJ cana-1219 36 2 βˆ’ βˆ’ PROPN cana-1219 36 3 π‘π‘œπ‘™π‘œπ‘Ÿπ‘Žπ‘π‘™π‘’ π‘π‘œπ‘™π‘œπ‘Ÿπ‘Žπ‘π‘™π‘’ VERB cana-1219 36 4 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN cana-1219 36 5 π‘˜ π‘˜ PROPN cana-1219 37 1 + + NOUN cana-1219 37 2 𝑙 𝑙 NOUN cana-1219 37 3 + + NUM cana-1219 37 4 π‘Ÿ π‘Ÿ NOUN cana-1219 37 5 = = SYM cana-1219 37 6 π‘ž π‘ž X cana-1219 37 7 } } PUNCT cana-1219 37 8 3 3 NUM cana-1219 37 9 . . X cana-1219 37 10 main main ADJ cana-1219 37 11 results result NOUN cana-1219 37 12 : : PUNCT cana-1219 37 13 3.1 3.1 NUM cana-1219 37 14 : : PUNCT cana-1219 37 15 star star NOUN cana-1219 37 16 cochromatic cochromatic ADJ cana-1219 37 17 number number NOUN cana-1219 37 18 of of ADP cana-1219 37 19 prism prism NOUN cana-1219 37 20 graphs graph NOUN cana-1219 37 21 theorem theorem VERB cana-1219 37 22 3.1.1 3.1.1 NUM cana-1219 37 23 : : PUNCT cana-1219 37 24 the the DET cana-1219 37 25 star star NOUN cana-1219 37 26 cochromatic cochromatic ADJ cana-1219 37 27 number number NOUN cana-1219 37 28 of of ADP cana-1219 37 29 prism prism NOUN cana-1219 37 30 graph graph NOUN cana-1219 37 31 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š PRON cana-1219 37 32 ] ] PUNCT cana-1219 37 33 for for ADP cana-1219 37 34 π‘š π‘š PROPN cana-1219 37 35 β‰₯ β‰₯ NUM cana-1219 37 36 3 3 NUM cana-1219 37 37 is be AUX cana-1219 37 38 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š SYM cana-1219 37 39 ] ] X cana-1219 37 40 = = SYM cana-1219 37 41 ⌈ ⌈ NOUN cana-1219 37 42 π‘š π‘š ADP cana-1219 37 43 2 2 NUM cana-1219 37 44 βŒ‰ βŒ‰ NOUN cana-1219 37 45 proof proof NOUN cana-1219 37 46 : : PUNCT cana-1219 37 47 let let VERB cana-1219 37 48 π‘Œπ‘š π‘Œπ‘š PROPN cana-1219 37 49 be be AUX cana-1219 37 50 prism prism NOUN cana-1219 37 51 graph graph NOUN cana-1219 37 52 with with ADP cana-1219 37 53 2 2 NUM cana-1219 37 54 m m NOUN cana-1219 37 55 vertices vertex NOUN cana-1219 37 56 and and CCONJ cana-1219 37 57 3 3 NUM cana-1219 37 58 m m NOUN cana-1219 37 59 edges edge NOUN cana-1219 37 60 and and CCONJ cana-1219 37 61 𝑉[π‘Œπ‘š 𝑉[π‘Œπ‘š NOUN cana-1219 37 62 ] ] X cana-1219 37 63 = = PUNCT cana-1219 37 64 { { PUNCT cana-1219 37 65 𝑣𝑛 𝑣𝑛 NOUN cana-1219 37 66 : : SYM cana-1219 37 67 1 1 NUM cana-1219 37 68 ≀ ≀ NUM cana-1219 37 69 𝑛 𝑛 PRON cana-1219 37 70 ≀ ≀ NUM cana-1219 37 71 π‘š π‘š NOUN cana-1219 37 72 } } PUNCT cana-1219 37 73 βˆͺ βˆͺ ADJ cana-1219 37 74 { { PUNCT cana-1219 37 75 𝑒𝑛 𝑒𝑛 NOUN cana-1219 37 76 : : SYM cana-1219 37 77 1 1 NUM cana-1219 37 78 ≀ ≀ NUM cana-1219 37 79 𝑛 𝑛 DET cana-1219 37 80 ≀ ≀ NUM cana-1219 37 81 π‘š π‘š NOUN cana-1219 37 82 } } PUNCT cana-1219 37 83 . . PUNCT cana-1219 38 1 consider consider VERB cana-1219 38 2 the the DET cana-1219 38 3 color color NOUN cana-1219 38 4 class class NOUN cana-1219 38 5 𝐢 𝐢 PROPN cana-1219 38 6 = = PRON cana-1219 38 7 { { PUNCT cana-1219 38 8 𝑐1 𝑐1 NOUN cana-1219 38 9 , , PUNCT cana-1219 38 10 𝑐2 𝑐2 NOUN cana-1219 38 11 , , PUNCT cana-1219 38 12 𝑐3 𝑐3 NOUN cana-1219 38 13 , , PUNCT cana-1219 38 14 … … PUNCT cana-1219 38 15 , , PUNCT cana-1219 38 16 𝑐 𝑐 PROPN cana-1219 38 17 ⌈ ⌈ NOUN cana-1219 38 18 π‘š π‘š ADP cana-1219 38 19 2 2 NUM cana-1219 38 20 βŒ‰ βŒ‰ NOUN cana-1219 38 21 } } PUNCT cana-1219 38 22 . . PUNCT cana-1219 39 1 define define VERB cana-1219 39 2 mapping mapping NOUN cana-1219 39 3 𝜎 𝜎 NOUN cana-1219 39 4 : : PUNCT cana-1219 39 5 { { PUNCT cana-1219 39 6 𝑣𝑛 𝑣𝑛 NOUN cana-1219 39 7 : : SYM cana-1219 39 8 1 1 NUM cana-1219 39 9 ≀ ≀ NUM cana-1219 39 10 𝑛 𝑛 PRON cana-1219 39 11 ≀ ≀ NUM cana-1219 39 12 π‘š π‘š NOUN cana-1219 39 13 } } PUNCT cana-1219 39 14 βˆͺ βˆͺ ADJ cana-1219 39 15 { { PUNCT cana-1219 39 16 𝑒𝑛 𝑒𝑛 NOUN cana-1219 39 17 : : SYM cana-1219 39 18 1 1 NUM cana-1219 39 19 ≀ ≀ NUM cana-1219 39 20 𝑛 𝑛 DET cana-1219 39 21 ≀ ≀ NUM cana-1219 39 22 π‘š π‘š ADP cana-1219 39 23 } } PUNCT cana-1219 39 24 β†’ β†’ SYM cana-1219 39 25 π‘π‘˜ π‘π‘˜ NOUN cana-1219 39 26 βˆ€ βˆ€ X cana-1219 39 27 π‘˜ π‘˜ X cana-1219 39 28 = = NOUN cana-1219 39 29 1,2,3 1,2,3 NUM cana-1219 39 30 , , PUNCT cana-1219 39 31 … … PUNCT cana-1219 39 32 case-(i case-(i PROPN cana-1219 39 33 ): ): PUNCT cana-1219 39 34 π’Ž π’Ž PROPN cana-1219 39 35 ≑ ≑ PROPN cana-1219 39 36 𝟎 𝟎 PROPN cana-1219 39 37 π’Žπ’π’… π’Žπ’π’… VERB cana-1219 39 38 πŸ’ πŸ’ NUM cana-1219 39 39 β€’ β€’ NOUN cana-1219 39 40 𝜎(𝑒4π‘˜ 𝜎(𝑒4π‘˜ NOUN cana-1219 39 41 , , PUNCT cana-1219 39 42 𝑒4π‘˜βˆ’1 𝑒4π‘˜βˆ’1 PROPN cana-1219 39 43 , , PUNCT cana-1219 39 44 𝑒4π‘˜+1 𝑒4π‘˜+1 NOUN cana-1219 39 45 , , PUNCT cana-1219 39 46 𝑣4π‘˜ 𝑣4π‘˜ NOUN cana-1219 39 47 ) ) PUNCT cana-1219 39 48 = = PUNCT cana-1219 40 1 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 40 2 β€’ β€’ NUM cana-1219 40 3 𝜎(𝑣4π‘˜βˆ’2 𝜎(𝑣4π‘˜βˆ’2 PROPN cana-1219 40 4 , , PUNCT cana-1219 40 5 𝑣4π‘˜βˆ’3 𝑣4π‘˜βˆ’3 PROPN cana-1219 40 6 , , PUNCT cana-1219 40 7 𝑣4π‘˜βˆ’1 𝑣4π‘˜βˆ’1 PROPN cana-1219 40 8 , , PUNCT cana-1219 40 9 𝑒4π‘˜βˆ’2 𝑒4π‘˜βˆ’2 NOUN cana-1219 40 10 ) ) PUNCT cana-1219 40 11 = = SYM cana-1219 41 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 41 2 case-(ii case-(ii PROPN cana-1219 41 3 ): ): PUNCT cana-1219 41 4 π’Ž π’Ž PROPN cana-1219 41 5 ≑ ≑ PROPN cana-1219 41 6 𝟏 𝟏 NUM cana-1219 41 7 π’Žπ’π’… π’Žπ’π’… VERB cana-1219 41 8 πŸ’ πŸ’ NUM cana-1219 41 9 β€’ β€’ NOUN cana-1219 41 10 𝜎(𝑒4π‘˜βˆ’2 𝜎(𝑒4π‘˜βˆ’2 PROPN cana-1219 41 11 , , PUNCT cana-1219 41 12 𝑒4π‘˜βˆ’1 𝑒4π‘˜βˆ’1 PROPN cana-1219 41 13 , , PUNCT cana-1219 41 14 𝑒4π‘˜βˆ’3 𝑒4π‘˜βˆ’3 PROPN cana-1219 41 15 , , PUNCT cana-1219 41 16 𝑣4π‘˜βˆ’2 𝑣4π‘˜βˆ’2 PROPN cana-1219 41 17 ) ) PUNCT cana-1219 41 18 = = SYM cana-1219 42 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 42 2 β€’ β€’ NUM cana-1219 42 3 𝜎(𝑣4π‘˜ 𝜎(𝑣4π‘˜ NOUN cana-1219 42 4 , , PUNCT cana-1219 42 5 𝑣4π‘˜+1 𝑣4π‘˜+1 PROPN cana-1219 42 6 , , PUNCT cana-1219 42 7 𝑣4π‘˜βˆ’1 𝑣4π‘˜βˆ’1 PROPN cana-1219 42 8 , , PUNCT cana-1219 42 9 𝑒4π‘˜ 𝑒4π‘˜ NOUN cana-1219 42 10 ) ) PUNCT cana-1219 42 11 = = PUNCT cana-1219 43 1 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 43 2 β€’ β€’ NUM cana-1219 43 3 𝜎(𝑣1 𝜎(𝑣1 X cana-1219 43 4 , , PUNCT cana-1219 43 5 π‘’π‘š π‘’π‘š NOUN cana-1219 43 6 ) ) PUNCT cana-1219 43 7 = = SYM cana-1219 43 8 𝑐 𝑐 PROPN cana-1219 43 9 ⌈ ⌈ X cana-1219 43 10 π‘š π‘š ADP cana-1219 43 11 2 2 NUM cana-1219 43 12 βŒ‰ βŒ‰ X cana-1219 43 13 case-(iii case-(iii NOUN cana-1219 43 14 ): ): PUNCT cana-1219 43 15 π’Ž π’Ž PROPN cana-1219 43 16 ≑ ≑ PROPN cana-1219 43 17 𝟐 𝟐 NUM cana-1219 43 18 π’Žπ’π’… π’Žπ’π’… NOUN cana-1219 43 19 πŸ’ πŸ’ NUM cana-1219 43 20 assign assign VERB cana-1219 43 21 the the DET cana-1219 43 22 coloring coloring NOUN cana-1219 43 23 as as SCONJ cana-1219 43 24 follows follow VERB cana-1219 43 25 : : PUNCT cana-1219 43 26 β€’ β€’ NUM cana-1219 43 27 𝜎(𝑒4π‘˜+1 𝜎(𝑒4π‘˜+1 NUM cana-1219 43 28 , , PUNCT cana-1219 43 29 𝑒4π‘˜ 𝑒4π‘˜ NOUN cana-1219 43 30 , , PUNCT cana-1219 43 31 𝑒4π‘˜+2 𝑒4π‘˜+2 PROPN cana-1219 43 32 , , PUNCT cana-1219 43 33 𝑣4π‘˜+1 𝑣4π‘˜+1 NOUN cana-1219 43 34 ) ) PUNCT cana-1219 43 35 = = PUNCT cana-1219 43 36 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 43 37 β€’ β€’ NOUN cana-1219 43 38 𝜎(𝑣4π‘˜+3 𝜎(𝑣4π‘˜+3 NUM cana-1219 43 39 , , PUNCT cana-1219 43 40 𝑣4π‘˜+2 𝑣4π‘˜+2 NOUN cana-1219 43 41 , , PUNCT cana-1219 43 42 𝑣4π‘˜+4 𝑣4π‘˜+4 NOUN cana-1219 43 43 , , PUNCT cana-1219 43 44 𝑒4π‘˜+3 𝑒4π‘˜+3 ADJ cana-1219 43 45 ) ) PUNCT cana-1219 44 1 = = SYM cana-1219 44 2 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 44 3 communications communication NOUN cana-1219 44 4 on on ADP cana-1219 44 5 applied apply VERB cana-1219 44 6 nonlinear nonlinear ADJ cana-1219 44 7 analysis analysis NOUN cana-1219 44 8 issn issn NOUN cana-1219 44 9 : : PUNCT cana-1219 44 10 1074 1074 NUM cana-1219 44 11 - - PUNCT cana-1219 44 12 133x 133x NUM cana-1219 44 13 vol vol NOUN cana-1219 44 14 31 31 NUM cana-1219 44 15 no no NOUN cana-1219 44 16 . . PUNCT cana-1219 45 1 6s 6s NUM cana-1219 45 2 ( ( PUNCT cana-1219 45 3 2024 2024 NUM cana-1219 45 4 ) ) PUNCT cana-1219 45 5 254 254 NUM cana-1219 45 6 https://internationalpubls.com https://internationalpubls.com X cana-1219 45 7 there there PRON cana-1219 45 8 exists exist VERB cana-1219 45 9 an an DET cana-1219 45 10 uncolored uncolored ADJ cana-1219 45 11 set set NOUN cana-1219 45 12 of of ADP cana-1219 45 13 vertices vertex NOUN cana-1219 45 14 { { PUNCT cana-1219 45 15 𝑣1 𝑣1 PROPN cana-1219 45 16 , , PUNCT cana-1219 45 17 𝑣2 𝑣2 PROPN cana-1219 45 18 , , PUNCT cana-1219 45 19 𝑣3 𝑣3 PROPN cana-1219 45 20 , , PUNCT cana-1219 45 21 𝑣4 𝑣4 NOUN cana-1219 45 22 , , PUNCT cana-1219 45 23 π‘£π‘š π‘£π‘š ADJ cana-1219 45 24 , , PUNCT cana-1219 45 25 𝑒1 𝑒1 PROPN cana-1219 45 26 , , PUNCT cana-1219 45 27 𝑒2 𝑒2 PROPN cana-1219 45 28 , , PUNCT cana-1219 45 29 𝑒3 𝑒3 PROPN cana-1219 45 30 } } PUNCT cana-1219 45 31 . . PUNCT cana-1219 46 1 in in ADP cana-1219 46 2 which which PRON cana-1219 46 3 the the DET cana-1219 46 4 vertices vertex NOUN cana-1219 46 5 { { PUNCT cana-1219 46 6 𝑣2 𝑣2 NOUN cana-1219 46 7 , , PUNCT cana-1219 46 8 𝑣1 𝑣1 NOUN cana-1219 46 9 , , PUNCT cana-1219 46 10 𝑣3 𝑣3 ADJ cana-1219 46 11 , , PUNCT cana-1219 46 12 𝑒2 𝑒2 NOUN cana-1219 46 13 } } PUNCT cana-1219 46 14 forms form VERB cana-1219 46 15 π‘˜1,3 π‘˜1,3 NOUN cana-1219 46 16 which which PRON cana-1219 46 17 is be AUX cana-1219 46 18 to to PART cana-1219 46 19 be be AUX cana-1219 46 20 colored color VERB cana-1219 46 21 with with ADP cana-1219 46 22 𝑐 𝑐 PROPN cana-1219 46 23 ⌈ ⌈ NOUN cana-1219 46 24 π‘š π‘š ADP cana-1219 46 25 2 2 NUM cana-1219 46 26 βŒ‰βˆ’1 βŒ‰βˆ’1 NOUN cana-1219 46 27 and and CCONJ cana-1219 46 28 the the DET cana-1219 46 29 vertices vertex NOUN cana-1219 46 30 { { PUNCT cana-1219 46 31 𝑒1 𝑒1 PROPN cana-1219 46 32 , , PUNCT cana-1219 46 33 𝑒3 𝑒3 PROPN cana-1219 46 34 , , PUNCT cana-1219 46 35 𝑣4 𝑣4 NOUN cana-1219 46 36 , , PUNCT cana-1219 46 37 π‘’π‘š π‘’π‘š ADP cana-1219 46 38 } } PUNCT cana-1219 46 39 forms form VERB cana-1219 46 40 an an DET cana-1219 46 41 independent independent ADJ cana-1219 46 42 set set NOUN cana-1219 46 43 is be AUX cana-1219 46 44 colored color VERB cana-1219 46 45 with with ADP cana-1219 46 46 new new ADJ cana-1219 46 47 color color NOUN cana-1219 46 48 𝑐 𝑐 PROPN cana-1219 46 49 ⌈ ⌈ NOUN cana-1219 46 50 π‘š π‘š ADP cana-1219 46 51 2 2 NUM cana-1219 46 52 βŒ‰ βŒ‰ X cana-1219 46 53 case-(iv case-(iv PROPN cana-1219 46 54 ): ): PUNCT cana-1219 46 55 π’Ž π’Ž PROPN cana-1219 46 56 ≑ ≑ PROPN cana-1219 46 57 πŸ‘ πŸ‘ NUM cana-1219 46 58 π’Žπ’π’… π’Žπ’π’… NOUN cana-1219 46 59 πŸ’ πŸ’ NUM cana-1219 46 60 assign assign VERB cana-1219 46 61 the the DET cana-1219 46 62 coloring coloring NOUN cana-1219 46 63 as as SCONJ cana-1219 46 64 follows follow VERB cana-1219 46 65 : : PUNCT cana-1219 46 66 β€’ β€’ NUM cana-1219 46 67 𝜎(𝑒4π‘˜ 𝜎(𝑒4π‘˜ NOUN cana-1219 46 68 , , PUNCT cana-1219 46 69 𝑒4π‘˜βˆ’1 𝑒4π‘˜βˆ’1 PROPN cana-1219 46 70 , , PUNCT cana-1219 46 71 𝑒4π‘˜+1 𝑒4π‘˜+1 NOUN cana-1219 46 72 , , PUNCT cana-1219 46 73 𝑣4π‘˜ 𝑣4π‘˜ NOUN cana-1219 46 74 ) ) PUNCT cana-1219 46 75 = = PUNCT cana-1219 47 1 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 47 2 β€’ β€’ NUM cana-1219 47 3 𝜎(𝑣4π‘˜βˆ’2 𝜎(𝑣4π‘˜βˆ’2 PROPN cana-1219 47 4 , , PUNCT cana-1219 47 5 𝑣4π‘˜βˆ’3 𝑣4π‘˜βˆ’3 PROPN cana-1219 47 6 , , PUNCT cana-1219 47 7 𝑣4π‘˜βˆ’1 𝑣4π‘˜βˆ’1 PROPN cana-1219 47 8 , , PUNCT cana-1219 47 9 𝑒4π‘˜βˆ’2 𝑒4π‘˜βˆ’2 NOUN cana-1219 47 10 ) ) PUNCT cana-1219 47 11 = = SYM cana-1219 48 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 48 2 β€’ β€’ NUM cana-1219 48 3 𝜎(𝑒1 𝜎(𝑒1 NOUN cana-1219 48 4 , , PUNCT cana-1219 48 5 π‘’π‘š π‘’π‘š NOUN cana-1219 48 6 ) ) PUNCT cana-1219 48 7 = = SYM cana-1219 48 8 𝑐 𝑐 PROPN cana-1219 48 9 ⌈ ⌈ X cana-1219 48 10 π‘š π‘š ADP cana-1219 48 11 2 2 NUM cana-1219 48 12 βŒ‰ βŒ‰ X cana-1219 48 13 to to PART cana-1219 48 14 prove prove VERB cana-1219 48 15 : : PUNCT cana-1219 48 16 π‘§βˆ—[π‘Œπ‘š π‘§βˆ—[π‘Œπ‘š PROPN cana-1219 48 17 ] ] PUNCT cana-1219 48 18 < < X cana-1219 48 19 ⌈ ⌈ X cana-1219 48 20 π‘š π‘š ADP cana-1219 48 21 2 2 NUM cana-1219 48 22 βŒ‰ βŒ‰ NOUN cana-1219 48 23 ; ; PUNCT cana-1219 48 24 π‘§βˆ—[π‘Œπ‘š π‘§βˆ—[π‘Œπ‘š PROPN cana-1219 48 25 ] ] PUNCT cana-1219 48 26 exists exist VERB cana-1219 48 27 ( ( PUNCT cana-1219 48 28 π‘š π‘š PROPN cana-1219 48 29 βˆ’ βˆ’ NUM cana-1219 48 30 12 12 NUM cana-1219 48 31 ) ) PUNCT cana-1219 48 32 βˆ’ βˆ’ NOUN cana-1219 48 33 𝐾1,3 𝐾1,3 NOUN cana-1219 48 34 stars star NOUN cana-1219 48 35 , , PUNCT cana-1219 48 36 each each DET cana-1219 48 37 color color NOUN cana-1219 48 38 class class NOUN cana-1219 48 39 is be AUX cana-1219 48 40 colored color VERB cana-1219 48 41 with with ADP cana-1219 48 42 one one NUM cana-1219 48 43 color color NOUN cana-1219 48 44 . . PUNCT cana-1219 49 1 hence hence ADV cana-1219 49 2 , , PUNCT cana-1219 49 3 there there PRON cana-1219 49 4 exists exist VERB cana-1219 49 5 an an DET cana-1219 49 6 independent independent ADJ cana-1219 49 7 set set NOUN cana-1219 49 8 remains remain VERB cana-1219 49 9 uncolored uncolored ADJ cana-1219 49 10 we we PRON cana-1219 49 11 need need VERB cana-1219 49 12 one one NUM cana-1219 49 13 more more ADJ cana-1219 49 14 color color NOUN cana-1219 49 15 to to PART cana-1219 49 16 complete complete VERB cana-1219 49 17 the the DET cana-1219 49 18 graph graph NOUN cana-1219 49 19 . . PUNCT cana-1219 50 1 therefore therefore ADV cana-1219 50 2 , , PUNCT cana-1219 50 3 assumption assumption NOUN cana-1219 50 4 is be AUX cana-1219 50 5 contradictory contradictory ADJ cana-1219 50 6 . . PUNCT cana-1219 51 1 hence hence ADV cana-1219 51 2 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š ADP cana-1219 51 3 ] ] X cana-1219 51 4 = = SYM cana-1219 51 5 ⌈ ⌈ NOUN cana-1219 51 6 π‘š π‘š ADP cana-1219 51 7 2 2 NUM cana-1219 51 8 βŒ‰ βŒ‰ NOUN cana-1219 51 9 . . PUNCT cana-1219 52 1 3.2 3.2 NUM cana-1219 52 2 : : PUNCT cana-1219 52 3 star star NOUN cana-1219 52 4 cochromatic cochromatic ADJ cana-1219 52 5 number number NOUN cana-1219 52 6 of of ADP cana-1219 52 7 line line NOUN cana-1219 52 8 graph graph NOUN cana-1219 52 9 of of ADP cana-1219 52 10 prism prism NOUN cana-1219 52 11 graphs graph NOUN cana-1219 52 12 theorem theorem VERB cana-1219 52 13 3.2.1 3.2.1 NUM cana-1219 52 14 : : PUNCT cana-1219 52 15 the the DET cana-1219 52 16 star star NOUN cana-1219 52 17 chromatic chromatic ADJ cana-1219 52 18 number number NOUN cana-1219 52 19 of of ADP cana-1219 52 20 line line NOUN cana-1219 52 21 graph graph NOUN cana-1219 52 22 of of ADP cana-1219 52 23 prism prism NOUN cana-1219 52 24 graph graph NOUN cana-1219 52 25 π‘ž[𝐿(π‘Œπ‘š π‘ž[𝐿(π‘Œπ‘š PROPN cana-1219 52 26 ) ) PUNCT cana-1219 52 27 ] ] PUNCT cana-1219 52 28 for for ADP cana-1219 52 29 π‘š π‘š PROPN cana-1219 52 30 β‰₯ β‰₯ NUM cana-1219 52 31 3 3 NUM cana-1219 52 32 is be AUX cana-1219 52 33 π‘ž[𝐿(π‘Œπ‘š π‘ž[𝐿(π‘Œπ‘š NOUN cana-1219 52 34 ) ) PUNCT cana-1219 52 35 ] ] PUNCT cana-1219 53 1 = = PUNCT cana-1219 53 2 ⌈ ⌈ X cana-1219 53 3 π‘š+2 π‘š+2 X cana-1219 53 4 2 2 NUM cana-1219 53 5 βŒ‰ βŒ‰ DET cana-1219 53 6 proof proof NOUN cana-1219 53 7 : : PUNCT cana-1219 53 8 let let VERB cana-1219 53 9 π‘Œπ‘š π‘Œπ‘š PROPN cana-1219 53 10 be be AUX cana-1219 53 11 prism prism NOUN cana-1219 53 12 graph graph NOUN cana-1219 53 13 with with ADP cana-1219 53 14 2 2 NUM cana-1219 53 15 m m NOUN cana-1219 53 16 vertices vertex NOUN cana-1219 53 17 and and CCONJ cana-1219 53 18 3 3 NUM cana-1219 53 19 m m NOUN cana-1219 53 20 edges edge NOUN cana-1219 53 21 and and CCONJ cana-1219 53 22 𝑉[π‘Œπ‘š 𝑉[π‘Œπ‘š NOUN cana-1219 53 23 ] ] X cana-1219 53 24 = = PUNCT cana-1219 53 25 { { PUNCT cana-1219 53 26 𝑣𝑛 𝑣𝑛 NOUN cana-1219 53 27 : : SYM cana-1219 53 28 1 1 NUM cana-1219 53 29 ≀ ≀ NUM cana-1219 53 30 𝑛 𝑛 PRON cana-1219 53 31 ≀ ≀ NUM cana-1219 53 32 π‘š π‘š NOUN cana-1219 53 33 } } PUNCT cana-1219 53 34 βˆͺ βˆͺ ADJ cana-1219 53 35 { { PUNCT cana-1219 53 36 𝑒𝑛 𝑒𝑛 NOUN cana-1219 53 37 : : SYM cana-1219 53 38 1 1 NUM cana-1219 53 39 ≀ ≀ NUM cana-1219 53 40 𝑛 𝑛 DET cana-1219 53 41 ≀ ≀ NUM cana-1219 53 42 π‘š π‘š NOUN cana-1219 53 43 } } PUNCT cana-1219 53 44 and and CCONJ cana-1219 53 45 [ [ X cana-1219 53 46 𝐿(π‘Œπ‘š 𝐿(π‘Œπ‘š NOUN cana-1219 53 47 ) ) PUNCT cana-1219 53 48 ] ] PUNCT cana-1219 53 49 be be AUX cana-1219 53 50 line line NOUN cana-1219 53 51 graph graph NOUN cana-1219 53 52 of of ADP cana-1219 53 53 prism prism NOUN cana-1219 53 54 graph graph NOUN cana-1219 53 55 with with ADP cana-1219 53 56 3 3 NUM cana-1219 53 57 m m NOUN cana-1219 53 58 vertices vertex NOUN cana-1219 53 59 and and CCONJ cana-1219 53 60 6 6 NUM cana-1219 53 61 m m NOUN cana-1219 53 62 edges edge NOUN cana-1219 53 63 where where SCONJ cana-1219 53 64 𝑉[𝐿(π‘Œπ‘š 𝑉[𝐿(π‘Œπ‘š NOUN cana-1219 53 65 ) ) PUNCT cana-1219 53 66 ] ] PUNCT cana-1219 54 1 = = PUNCT cana-1219 54 2 { { PUNCT cana-1219 54 3 𝑣𝑛 𝑣𝑛 NOUN cana-1219 54 4 : : SYM cana-1219 54 5 1 1 NUM cana-1219 54 6 ≀ ≀ NUM cana-1219 54 7 𝑛 𝑛 PRON cana-1219 54 8 ≀ ≀ NUM cana-1219 54 9 π‘š π‘š NOUN cana-1219 54 10 } } PUNCT cana-1219 54 11 βˆͺ βˆͺ ADJ cana-1219 54 12 { { PUNCT cana-1219 54 13 𝑒𝑛 𝑒𝑛 NOUN cana-1219 54 14 : : SYM cana-1219 54 15 1 1 NUM cana-1219 54 16 ≀ ≀ NUM cana-1219 54 17 𝑛 𝑛 PRON cana-1219 54 18 ≀ ≀ NUM cana-1219 54 19 π‘š π‘š NOUN cana-1219 54 20 } } PUNCT cana-1219 54 21 βˆͺ βˆͺ ADJ cana-1219 54 22 { { PUNCT cana-1219 54 23 𝑀𝑛 𝑀𝑛 NOUN cana-1219 54 24 : : PUNCT cana-1219 54 25 1 1 NUM cana-1219 54 26 ≀ ≀ NUM cana-1219 54 27 𝑛 𝑛 DET cana-1219 54 28 ≀ ≀ NUM cana-1219 54 29 π‘š π‘š NOUN cana-1219 54 30 } } PUNCT cana-1219 54 31 . . PUNCT cana-1219 55 1 consider consider VERB cana-1219 55 2 the the DET cana-1219 55 3 color color NOUN cana-1219 55 4 class class NOUN cana-1219 55 5 𝐢 𝐢 PROPN cana-1219 55 6 = = PRON cana-1219 55 7 { { PUNCT cana-1219 55 8 𝑐1 𝑐1 NOUN cana-1219 55 9 , , PUNCT cana-1219 55 10 𝑐2 𝑐2 NOUN cana-1219 55 11 , , PUNCT cana-1219 55 12 𝑐3 𝑐3 NOUN cana-1219 55 13 , , PUNCT cana-1219 55 14 … … PUNCT cana-1219 55 15 , , PUNCT cana-1219 55 16 𝑐 𝑐 PROPN cana-1219 55 17 ⌈ ⌈ NOUN cana-1219 55 18 π‘š+2 π‘š+2 X cana-1219 55 19 2 2 NUM cana-1219 55 20 βŒ‰ βŒ‰ X cana-1219 55 21 } } PUNCT cana-1219 55 22 . . PUNCT cana-1219 56 1 define define VERB cana-1219 56 2 mapping mapping NOUN cana-1219 56 3 𝜎 𝜎 NOUN cana-1219 56 4 : : PUNCT cana-1219 56 5 { { PUNCT cana-1219 56 6 𝑣𝑛 𝑣𝑛 NOUN cana-1219 56 7 : : SYM cana-1219 56 8 1 1 NUM cana-1219 56 9 ≀ ≀ NUM cana-1219 56 10 𝑛 𝑛 PRON cana-1219 56 11 ≀ ≀ NUM cana-1219 56 12 π‘š π‘š NOUN cana-1219 56 13 } } PUNCT cana-1219 56 14 βˆͺ βˆͺ ADJ cana-1219 56 15 { { PUNCT cana-1219 56 16 𝑒𝑛 𝑒𝑛 NOUN cana-1219 56 17 : : SYM cana-1219 56 18 1 1 NUM cana-1219 56 19 ≀ ≀ NUM cana-1219 56 20 𝑛 𝑛 PRON cana-1219 56 21 ≀ ≀ NUM cana-1219 56 22 π‘š π‘š NOUN cana-1219 56 23 } } PUNCT cana-1219 56 24 βˆͺ βˆͺ ADJ cana-1219 56 25 { { PUNCT cana-1219 56 26 𝑀𝑛 𝑀𝑛 NOUN cana-1219 56 27 : : PUNCT cana-1219 56 28 1 1 NUM cana-1219 56 29 ≀ ≀ NUM cana-1219 56 30 𝑛 𝑛 DET cana-1219 56 31 ≀ ≀ NUM cana-1219 56 32 π‘š π‘š ADP cana-1219 56 33 } } PUNCT cana-1219 56 34 β†’ β†’ SYM cana-1219 56 35 π‘π‘˜ π‘π‘˜ NOUN cana-1219 56 36 βˆ€ βˆ€ X cana-1219 56 37 π‘˜ π‘˜ X cana-1219 56 38 = = NOUN cana-1219 56 39 1,2,3 1,2,3 NUM cana-1219 56 40 , , PUNCT cana-1219 56 41 … … PUNCT cana-1219 56 42 case-(i case-(i PROPN cana-1219 56 43 ): ): PUNCT cana-1219 56 44 π’Ž π’Ž X cana-1219 56 45 = = PUNCT cana-1219 56 46 𝒆𝒗𝒆𝒏 𝒆𝒗𝒆𝒏 NOUN cana-1219 56 47 assign assign VERB cana-1219 56 48 the the DET cana-1219 56 49 coloring coloring NOUN cana-1219 56 50 as as SCONJ cana-1219 56 51 follows follow VERB cana-1219 56 52 : : PUNCT cana-1219 56 53 β€’ β€’ NUM cana-1219 56 54 𝜎(𝑒4π‘˜βˆ’3 𝜎(𝑒4π‘˜βˆ’3 NOUN cana-1219 56 55 , , PUNCT cana-1219 56 56 𝑣4π‘˜βˆ’3 𝑣4π‘˜βˆ’3 PROPN cana-1219 56 57 , , PUNCT cana-1219 56 58 𝑣4π‘˜βˆ’2 𝑣4π‘˜βˆ’2 PROPN cana-1219 56 59 , , PUNCT cana-1219 56 60 𝑀4π‘˜βˆ’3 𝑀4π‘˜βˆ’3 PROPN cana-1219 56 61 , , PUNCT cana-1219 56 62 𝑀4π‘˜βˆ’2 𝑀4π‘˜βˆ’2 NOUN cana-1219 56 63 ) ) PUNCT cana-1219 56 64 = = SYM cana-1219 56 65 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 56 66 β€’ β€’ NUM cana-1219 56 67 𝜎(𝑒4π‘˜βˆ’1 𝜎(𝑒4π‘˜βˆ’1 PROPN cana-1219 56 68 , , PUNCT cana-1219 56 69 𝑣4π‘˜βˆ’1 𝑣4π‘˜βˆ’1 PROPN cana-1219 56 70 , , PUNCT cana-1219 56 71 𝑣4π‘˜ 𝑣4π‘˜ PRON cana-1219 56 72 , , PUNCT cana-1219 56 73 𝑀4π‘˜βˆ’1 𝑀4π‘˜βˆ’1 PROPN cana-1219 56 74 , , PUNCT cana-1219 56 75 𝑀4π‘˜ 𝑀4π‘˜ PROPN cana-1219 56 76 ) ) PUNCT cana-1219 57 1 = = PUNCT cana-1219 57 2 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 57 3 β€’ β€’ NUM cana-1219 57 4 𝜎(𝑒2π‘˜ 𝜎(𝑒2π‘˜ NOUN cana-1219 57 5 ) ) PUNCT cana-1219 57 6 = = SYM cana-1219 57 7 𝑐 𝑐 PROPN cana-1219 57 8 ⌈ ⌈ X cana-1219 57 9 π‘š+2 π‘š+2 X cana-1219 57 10 2 2 NUM cana-1219 57 11 βŒ‰ βŒ‰ SCONJ cana-1219 57 12 case-(ii case-(ii PROPN cana-1219 57 13 ): ): PUNCT cana-1219 57 14 π’Ž π’Ž PROPN cana-1219 57 15 = = SYM cana-1219 57 16 𝒐𝒅𝒅 𝒐𝒅𝒅 NOUN cana-1219 57 17 β€’ β€’ ADP cana-1219 57 18 𝜎(𝑒4π‘˜βˆ’2 𝜎(𝑒4π‘˜βˆ’2 NOUN cana-1219 57 19 , , PUNCT cana-1219 57 20 𝑣4π‘˜βˆ’3 𝑣4π‘˜βˆ’3 PROPN cana-1219 57 21 , , PUNCT cana-1219 57 22 𝑣4π‘˜βˆ’2 𝑣4π‘˜βˆ’2 PROPN cana-1219 57 23 , , PUNCT cana-1219 57 24 𝑀4π‘˜βˆ’3 𝑀4π‘˜βˆ’3 PROPN cana-1219 57 25 , , PUNCT cana-1219 57 26 𝑀4π‘˜βˆ’2 𝑀4π‘˜βˆ’2 NOUN cana-1219 57 27 ) ) PUNCT cana-1219 57 28 = = SYM cana-1219 57 29 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 57 30 β€’ β€’ NUM cana-1219 57 31 𝜎(𝑒4π‘˜ 𝜎(𝑒4π‘˜ PROPN cana-1219 57 32 , , PUNCT cana-1219 57 33 𝑣4π‘˜βˆ’1 𝑣4π‘˜βˆ’1 PROPN cana-1219 57 34 , , PUNCT cana-1219 57 35 𝑣4π‘˜ 𝑣4π‘˜ PRON cana-1219 57 36 , , PUNCT cana-1219 57 37 𝑀4π‘˜βˆ’1 𝑀4π‘˜βˆ’1 PROPN cana-1219 57 38 , , PUNCT cana-1219 57 39 𝑀4π‘˜ 𝑀4π‘˜ PROPN cana-1219 57 40 ) ) PUNCT cana-1219 57 41 = = PUNCT cana-1219 58 1 𝑐2π‘˜ 𝑐2π‘˜ NOUN cana-1219 58 2 β€’ β€’ NOUN cana-1219 58 3 𝜎(𝑒2π‘˜βˆ’1 𝜎(𝑒2π‘˜βˆ’1 NUM cana-1219 58 4 ) ) PUNCT cana-1219 59 1 = = SYM cana-1219 59 2 𝑐 𝑐 PROPN cana-1219 59 3 ⌈ ⌈ X cana-1219 59 4 π‘š+2 π‘š+2 X cana-1219 59 5 2 2 NUM cana-1219 59 6 βŒ‰βˆ’1 βŒ‰βˆ’1 PROPN cana-1219 59 7 β€’ β€’ NUM cana-1219 59 8 𝜎(π‘£π‘š 𝜎(π‘£π‘š PROPN cana-1219 59 9 , , PUNCT cana-1219 59 10 π‘€π‘š π‘€π‘š NOUN cana-1219 59 11 ) ) PUNCT cana-1219 59 12 = = SYM cana-1219 59 13 𝑐 𝑐 PROPN cana-1219 59 14 ⌈ ⌈ X cana-1219 59 15 π‘š+2 π‘š+2 X cana-1219 59 16 2 2 NUM cana-1219 59 17 βŒ‰ βŒ‰ PRON cana-1219 59 18 communications communication NOUN cana-1219 59 19 on on ADP cana-1219 59 20 applied apply VERB cana-1219 59 21 nonlinear nonlinear ADJ cana-1219 59 22 analysis analysis NOUN cana-1219 59 23 issn issn NOUN cana-1219 59 24 : : PUNCT cana-1219 59 25 1074 1074 NUM cana-1219 59 26 - - PUNCT cana-1219 59 27 133x 133x NUM cana-1219 59 28 vol vol NOUN cana-1219 59 29 31 31 NUM cana-1219 59 30 no no NOUN cana-1219 59 31 . . PUNCT cana-1219 60 1 6s 6s NUM cana-1219 60 2 ( ( PUNCT cana-1219 60 3 2024 2024 NUM cana-1219 60 4 ) ) PUNCT cana-1219 60 5 255 255 NUM cana-1219 60 6 https://internationalpubls.com https://internationalpubls.com X cana-1219 60 7 to to PART cana-1219 60 8 prove prove VERB cana-1219 60 9 : : PUNCT cana-1219 60 10 π‘§βˆ—[𝐿(π‘Œπ‘š π‘§βˆ—[𝐿(π‘Œπ‘š NOUN cana-1219 60 11 ) ) PUNCT cana-1219 60 12 ] ] PUNCT cana-1219 61 1 < < X cana-1219 61 2 ⌈ ⌈ X cana-1219 61 3 π‘š+2 π‘š+2 X cana-1219 61 4 2 2 NUM cana-1219 61 5 βŒ‰ βŒ‰ NOUN cana-1219 61 6 , , PUNCT cana-1219 61 7 say say VERB cana-1219 61 8 ⌈ ⌈ X cana-1219 61 9 π‘š+2 π‘š+2 X cana-1219 61 10 2 2 NUM cana-1219 61 11 βŒ‰ βŒ‰ PRON cana-1219 61 12 βˆ’ βˆ’ PROPN cana-1219 61 13 2 2 NUM cana-1219 61 14 ; ; PUNCT cana-1219 61 15 π‘§βˆ—[𝐿(π‘Œπ‘š π‘§βˆ—[𝐿(π‘Œπ‘š NOUN cana-1219 61 16 ) ) PUNCT cana-1219 61 17 ] ] PUNCT cana-1219 61 18 exists exist VERB cana-1219 61 19 ( ( PUNCT cana-1219 61 20 π‘š π‘š PROPN cana-1219 61 21 2 2 NUM cana-1219 61 22 ) ) PUNCT cana-1219 61 23 βˆ’ βˆ’ PROPN cana-1219 61 24 𝐾1,3 𝐾1,3 NOUN cana-1219 61 25 stars star NOUN cana-1219 61 26 , , PUNCT cana-1219 61 27 each each DET cana-1219 61 28 color color NOUN cana-1219 61 29 class class NOUN cana-1219 61 30 is be AUX cana-1219 61 31 colored color VERB cana-1219 61 32 with with ADP cana-1219 61 33 one one NUM cana-1219 61 34 color color NOUN cana-1219 61 35 . . PUNCT cana-1219 62 1 hence hence ADV cana-1219 62 2 , , PUNCT cana-1219 62 3 there there PRON cana-1219 62 4 exists exist VERB cana-1219 62 5 an an DET cana-1219 62 6 independent independent ADJ cana-1219 62 7 set set NOUN cana-1219 62 8 and and CCONJ cana-1219 62 9 clique clique NOUN cana-1219 62 10 remains remain VERB cana-1219 62 11 uncolored uncolored ADJ cana-1219 62 12 we we PRON cana-1219 62 13 need need VERB cana-1219 62 14 two two NUM cana-1219 62 15 more more ADJ cana-1219 62 16 colors color NOUN cana-1219 62 17 to to PART cana-1219 62 18 complete complete VERB cana-1219 62 19 the the DET cana-1219 62 20 graph graph NOUN cana-1219 62 21 . . PUNCT cana-1219 63 1 therefore therefore ADV cana-1219 63 2 , , PUNCT cana-1219 63 3 assumption assumption NOUN cana-1219 63 4 is be AUX cana-1219 63 5 contradictory contradictory ADJ cana-1219 63 6 . . PUNCT cana-1219 64 1 hence hence ADV cana-1219 64 2 π‘§βˆ—[𝐿(π‘Œπ‘š π‘§βˆ—[𝐿(π‘Œπ‘š NOUN cana-1219 64 3 ) ) PUNCT cana-1219 64 4 ] ] PUNCT cana-1219 65 1 = = PUNCT cana-1219 65 2 ⌈ ⌈ X cana-1219 65 3 π‘š+2 π‘š+2 X cana-1219 65 4 2 2 NUM cana-1219 65 5 βŒ‰. βŒ‰. ADV cana-1219 65 6 3.3 3.3 NUM cana-1219 65 7 : : PUNCT cana-1219 65 8 star star NOUN cana-1219 65 9 cochromatic cochromatic ADJ cana-1219 65 10 number number NOUN cana-1219 65 11 of of ADP cana-1219 65 12 middle middle ADJ cana-1219 65 13 graph graph NOUN cana-1219 65 14 of of ADP cana-1219 65 15 prism prism NOUN cana-1219 65 16 graphs graph NOUN cana-1219 65 17 theorem theorem VERB cana-1219 65 18 3.3.1 3.3.1 NUM cana-1219 65 19 : : PUNCT cana-1219 65 20 the the DET cana-1219 65 21 star star NOUN cana-1219 65 22 chromatic chromatic ADJ cana-1219 65 23 number number NOUN cana-1219 65 24 of of ADP cana-1219 65 25 middle middle ADJ cana-1219 65 26 graph graph NOUN cana-1219 65 27 of of ADP cana-1219 65 28 prism prism NOUN cana-1219 65 29 graph graph NOUN cana-1219 65 30 π‘ž[𝑀(π‘Œπ‘š π‘ž[𝑀(π‘Œπ‘š PROPN cana-1219 65 31 ) ) PUNCT cana-1219 65 32 ] ] PUNCT cana-1219 65 33 for for ADP cana-1219 65 34 π‘š π‘š PROPN cana-1219 65 35 ≑ ≑ PROPN cana-1219 65 36 0 0 NUM cana-1219 65 37 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-1219 65 38 2 2 NUM cana-1219 65 39 is be AUX cana-1219 65 40 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š SYM cana-1219 65 41 ] ] X cana-1219 65 42 = = SYM cana-1219 65 43 ⌈ ⌈ SYM cana-1219 65 44 π‘š+4 π‘š+4 NUM cana-1219 65 45 2 2 NUM cana-1219 65 46 βŒ‰ βŒ‰ DET cana-1219 65 47 proof proof NOUN cana-1219 65 48 : : PUNCT cana-1219 65 49 let let VERB cana-1219 65 50 π‘Œπ‘š π‘Œπ‘š PROPN cana-1219 65 51 be be AUX cana-1219 65 52 prism prism NOUN cana-1219 65 53 graph graph NOUN cana-1219 65 54 with with ADP cana-1219 65 55 2 2 NUM cana-1219 65 56 m m NOUN cana-1219 65 57 vertices vertex NOUN cana-1219 65 58 and and CCONJ cana-1219 65 59 3 3 NUM cana-1219 65 60 m m NOUN cana-1219 65 61 edges edge NOUN cana-1219 65 62 and and CCONJ cana-1219 65 63 𝑉[π‘Œπ‘š 𝑉[π‘Œπ‘š NOUN cana-1219 65 64 ] ] X cana-1219 65 65 = = PUNCT cana-1219 65 66 { { PUNCT cana-1219 65 67 𝑣𝑛 𝑣𝑛 NOUN cana-1219 65 68 : : SYM cana-1219 65 69 1 1 NUM cana-1219 65 70 ≀ ≀ NUM cana-1219 65 71 𝑛 𝑛 PRON cana-1219 65 72 ≀ ≀ NUM cana-1219 65 73 π‘š π‘š NOUN cana-1219 65 74 } } PUNCT cana-1219 65 75 βˆͺ βˆͺ ADJ cana-1219 65 76 { { PUNCT cana-1219 65 77 𝑒𝑛 𝑒𝑛 NOUN cana-1219 65 78 : : SYM cana-1219 65 79 1 1 NUM cana-1219 65 80 ≀ ≀ NUM cana-1219 65 81 𝑛 𝑛 DET cana-1219 65 82 ≀ ≀ NUM cana-1219 65 83 π‘š π‘š NOUN cana-1219 65 84 } } PUNCT cana-1219 65 85 and and CCONJ cana-1219 65 86 [ [ X cana-1219 65 87 𝑀(π‘Œπ‘š 𝑀(π‘Œπ‘š NOUN cana-1219 65 88 ) ) PUNCT cana-1219 65 89 ] ] PUNCT cana-1219 65 90 be be AUX cana-1219 65 91 middle middle ADJ cana-1219 65 92 graph graph NOUN cana-1219 65 93 of of ADP cana-1219 65 94 prism prism NOUN cana-1219 65 95 graph graph NOUN cana-1219 65 96 where where SCONJ cana-1219 65 97 subdividing subdivide VERB cana-1219 65 98 each each DET cana-1219 65 99 edge edge NOUN cana-1219 65 100 exactly exactly ADV cana-1219 65 101 once once ADV cana-1219 65 102 and and CCONJ cana-1219 65 103 join join VERB cana-1219 65 104 the the DET cana-1219 65 105 adjacent adjacent ADJ cana-1219 65 106 vertices vertex NOUN cana-1219 65 107 , , PUNCT cana-1219 65 108 the the DET cana-1219 65 109 vertex vertex NOUN cana-1219 65 110 set set NOUN cana-1219 65 111 of of ADP cana-1219 65 112 𝑉[𝑀(π‘Œπ‘š 𝑉[𝑀(π‘Œπ‘š PROPN cana-1219 65 113 ) ) PUNCT cana-1219 65 114 ] ] PUNCT cana-1219 66 1 = = PUNCT cana-1219 66 2 { { PUNCT cana-1219 66 3 𝑣𝑛 𝑣𝑛 NOUN cana-1219 66 4 : : SYM cana-1219 66 5 1 1 NUM cana-1219 66 6 ≀ ≀ NUM cana-1219 66 7 𝑛 𝑛 PRON cana-1219 66 8 ≀ ≀ NUM cana-1219 66 9 π‘š π‘š NOUN cana-1219 66 10 } } PUNCT cana-1219 66 11 βˆͺ βˆͺ ADJ cana-1219 66 12 { { PUNCT cana-1219 66 13 𝑣𝑛 𝑣𝑛 NOUN cana-1219 66 14 β€² β€² NUM cana-1219 66 15 : : PUNCT cana-1219 66 16 1 1 NUM cana-1219 66 17 ≀ ≀ NUM cana-1219 66 18 𝑛 𝑛 PRON cana-1219 66 19 ≀ ≀ NUM cana-1219 66 20 π‘š π‘š NOUN cana-1219 66 21 } } PUNCT cana-1219 66 22 βˆͺ βˆͺ ADJ cana-1219 66 23 { { PUNCT cana-1219 66 24 𝑒𝑛 𝑒𝑛 NOUN cana-1219 66 25 : : SYM cana-1219 66 26 1 1 NUM cana-1219 66 27 ≀ ≀ NUM cana-1219 66 28 𝑛 𝑛 PRON cana-1219 66 29 ≀ ≀ NUM cana-1219 66 30 π‘š π‘š NOUN cana-1219 66 31 } } PUNCT cana-1219 66 32 βˆͺ βˆͺ ADJ cana-1219 66 33 { { PUNCT cana-1219 66 34 𝑒𝑛 𝑒𝑛 NOUN cana-1219 66 35 β€² β€² NUM cana-1219 66 36 : : PUNCT cana-1219 66 37 1 1 NUM cana-1219 66 38 ≀ ≀ NUM cana-1219 66 39 𝑛 𝑛 PRON cana-1219 66 40 ≀ ≀ NUM cana-1219 66 41 π‘š π‘š NOUN cana-1219 66 42 } } PUNCT cana-1219 66 43 βˆͺ βˆͺ ADJ cana-1219 66 44 { { PUNCT cana-1219 66 45 𝑀𝑛 𝑀𝑛 NOUN cana-1219 66 46 β€² β€² NUM cana-1219 66 47 : : PUNCT cana-1219 66 48 1 1 NUM cana-1219 66 49 ≀ ≀ NUM cana-1219 66 50 𝑛 𝑛 DET cana-1219 66 51 ≀ ≀ NUM cana-1219 66 52 π‘š π‘š NOUN cana-1219 66 53 } } PUNCT cana-1219 66 54 . . PUNCT cana-1219 67 1 consider consider VERB cana-1219 67 2 the the DET cana-1219 67 3 color color NOUN cana-1219 67 4 class class NOUN cana-1219 67 5 𝐢 𝐢 PROPN cana-1219 67 6 = = PRON cana-1219 67 7 { { PUNCT cana-1219 67 8 𝑐1 𝑐1 NOUN cana-1219 67 9 , , PUNCT cana-1219 67 10 𝑐2 𝑐2 NOUN cana-1219 67 11 , , PUNCT cana-1219 67 12 𝑐3 𝑐3 NOUN cana-1219 67 13 , , PUNCT cana-1219 67 14 … … PUNCT cana-1219 67 15 , , PUNCT cana-1219 67 16 𝑐 𝑐 NOUN cana-1219 67 17 ⌈ ⌈ SYM cana-1219 67 18 π‘š+4 π‘š+4 NUM cana-1219 67 19 2 2 NUM cana-1219 67 20 βŒ‰ βŒ‰ NOUN cana-1219 67 21 } } PUNCT cana-1219 67 22 . . PUNCT cana-1219 68 1 define define VERB cana-1219 68 2 mapping mapping NOUN cana-1219 68 3 𝜎 𝜎 NOUN cana-1219 68 4 : : PUNCT cana-1219 68 5 { { PUNCT cana-1219 68 6 𝑣𝑛 𝑣𝑛 NOUN cana-1219 68 7 : : SYM cana-1219 68 8 1 1 NUM cana-1219 68 9 ≀ ≀ NUM cana-1219 68 10 𝑛 𝑛 PRON cana-1219 68 11 ≀ ≀ NUM cana-1219 68 12 π‘š π‘š NOUN cana-1219 68 13 } } PUNCT cana-1219 68 14 βˆͺ βˆͺ ADJ cana-1219 68 15 { { PUNCT cana-1219 68 16 𝑣𝑛 𝑣𝑛 NOUN cana-1219 68 17 β€² β€² NUM cana-1219 68 18 : : PUNCT cana-1219 68 19 1 1 NUM cana-1219 68 20 ≀ ≀ NUM cana-1219 68 21 𝑛 𝑛 PRON cana-1219 68 22 ≀ ≀ NUM cana-1219 68 23 π‘š π‘š NOUN cana-1219 68 24 } } PUNCT cana-1219 68 25 βˆͺ βˆͺ ADJ cana-1219 68 26 { { PUNCT cana-1219 68 27 𝑒𝑛 𝑒𝑛 NOUN cana-1219 68 28 : : SYM cana-1219 68 29 1 1 NUM cana-1219 68 30 ≀ ≀ NUM cana-1219 68 31 𝑛 𝑛 PRON cana-1219 68 32 ≀ ≀ NUM cana-1219 68 33 π‘š π‘š NOUN cana-1219 68 34 } } PUNCT cana-1219 68 35 βˆͺ βˆͺ ADJ cana-1219 68 36 { { PUNCT cana-1219 68 37 𝑒𝑛 𝑒𝑛 NOUN cana-1219 68 38 β€² β€² NUM cana-1219 68 39 : : PUNCT cana-1219 68 40 1 1 NUM cana-1219 68 41 ≀ ≀ NUM cana-1219 68 42 𝑛 𝑛 PRON cana-1219 68 43 ≀ ≀ NUM cana-1219 68 44 π‘š π‘š NOUN cana-1219 68 45 } } PUNCT cana-1219 68 46 βˆͺ βˆͺ ADJ cana-1219 68 47 { { PUNCT cana-1219 68 48 𝑀𝑛 𝑀𝑛 NOUN cana-1219 68 49 β€² β€² NUM cana-1219 68 50 : : PUNCT cana-1219 68 51 1 1 NUM cana-1219 68 52 ≀ ≀ NUM cana-1219 68 53 𝑛 𝑛 DET cana-1219 68 54 ≀ ≀ NUM cana-1219 68 55 π‘š π‘š ADP cana-1219 68 56 } } PUNCT cana-1219 68 57 β†’ β†’ SYM cana-1219 68 58 π‘π‘˜ π‘π‘˜ NOUN cana-1219 68 59 βˆ€ βˆ€ X cana-1219 68 60 π‘˜ π‘˜ X cana-1219 68 61 = = NOUN cana-1219 68 62 1,2,3 1,2,3 NUM cana-1219 68 63 , , PUNCT cana-1219 68 64 … … PUNCT cana-1219 68 65 assign assign VERB cana-1219 68 66 the the DET cana-1219 68 67 coloring coloring NOUN cana-1219 68 68 as as SCONJ cana-1219 68 69 follows follow VERB cana-1219 68 70 : : PUNCT cana-1219 68 71 β€’ β€’ NUM cana-1219 68 72 𝜎(𝑀4π‘˜βˆ’3 𝜎(𝑀4π‘˜βˆ’3 PROPN cana-1219 68 73 β€² β€² NOUN cana-1219 68 74 , , PUNCT cana-1219 68 75 𝑣4π‘˜βˆ’3 𝑣4π‘˜βˆ’3 NOUN cana-1219 68 76 β€² β€² NUM cana-1219 68 77 , , PUNCT cana-1219 68 78 𝑣4π‘˜βˆ’2 𝑣4π‘˜βˆ’2 PROPN cana-1219 68 79 β€² β€² NOUN cana-1219 68 80 , , PUNCT cana-1219 68 81 𝑒4π‘˜βˆ’3 𝑒4π‘˜βˆ’3 ADP cana-1219 68 82 β€² β€² NUM cana-1219 68 83 , , PUNCT cana-1219 68 84 𝑣4π‘˜βˆ’2 𝑣4π‘˜βˆ’2 PROPN cana-1219 68 85 β€² β€² NOUN cana-1219 68 86 , , PUNCT cana-1219 68 87 𝑣4π‘˜βˆ’2 𝑣4π‘˜βˆ’2 PROPN cana-1219 68 88 , , PUNCT cana-1219 68 89 𝑒4π‘˜βˆ’2 𝑒4π‘˜βˆ’2 NOUN cana-1219 68 90 ) ) PUNCT cana-1219 68 91 = = PUNCT cana-1219 69 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 69 2 β€’ β€’ NUM cana-1219 69 3 𝜎(𝑀4π‘˜βˆ’1 𝜎(𝑀4π‘˜βˆ’1 PROPN cana-1219 69 4 β€² β€² NUM cana-1219 69 5 , , PUNCT cana-1219 69 6 𝑣4π‘˜βˆ’1 𝑣4π‘˜βˆ’1 NUM cana-1219 69 7 β€² β€² NOUN cana-1219 69 8 , , PUNCT cana-1219 69 9 𝑣4π‘˜ 𝑣4π‘˜ ADJ cana-1219 69 10 β€² β€² NOUN cana-1219 69 11 , , PUNCT cana-1219 69 12 𝑒4π‘˜βˆ’1 𝑒4π‘˜βˆ’1 PROPN cana-1219 69 13 β€² β€² NUM cana-1219 69 14 , , PUNCT cana-1219 69 15 𝑒4π‘˜ 𝑒4π‘˜ NOUN cana-1219 69 16 β€² β€² NOUN cana-1219 69 17 , , PUNCT cana-1219 69 18 𝑒4π‘˜ 𝑒4π‘˜ NOUN cana-1219 69 19 , , PUNCT cana-1219 69 20 𝑣4π‘˜ 𝑣4π‘˜ NOUN cana-1219 69 21 ) ) PUNCT cana-1219 69 22 = = PUNCT cana-1219 70 1 𝑐2π‘˜ 𝑐2π‘˜ NOUN cana-1219 70 2 β€’ β€’ NOUN cana-1219 70 3 𝜎(𝑒2π‘˜βˆ’1 𝜎(𝑒2π‘˜βˆ’1 PROPN cana-1219 70 4 , , PUNCT cana-1219 70 5 𝑣2π‘˜βˆ’1 𝑣2π‘˜βˆ’1 PROPN cana-1219 70 6 ) ) PUNCT cana-1219 70 7 = = SYM cana-1219 71 1 𝑐 𝑐 PROPN cana-1219 71 2 ⌈ ⌈ X cana-1219 71 3 π‘š+4 π‘š+4 NOUN cana-1219 71 4 2 2 NUM cana-1219 71 5 βŒ‰βˆ’1 βŒ‰βˆ’1 PROPN cana-1219 71 6 β€’ β€’ NOUN cana-1219 71 7 𝜎(𝑀2π‘˜ 𝜎(𝑀2π‘˜ NOUN cana-1219 71 8 β€² β€² NUM cana-1219 71 9 ) ) PUNCT cana-1219 72 1 = = SYM cana-1219 72 2 𝑐 𝑐 PROPN cana-1219 72 3 ⌈ ⌈ X cana-1219 72 4 π‘š+4 π‘š+4 NUM cana-1219 72 5 2 2 NUM cana-1219 72 6 βŒ‰ βŒ‰ PRON cana-1219 72 7 to to PART cana-1219 72 8 prove prove VERB cana-1219 72 9 : : PUNCT cana-1219 72 10 π‘§βˆ—[𝑀(π‘Œπ‘š π‘§βˆ—[𝑀(π‘Œπ‘š ADJ cana-1219 72 11 ) ) PUNCT cana-1219 72 12 ] ] PUNCT cana-1219 73 1 < < X cana-1219 73 2 ⌈ ⌈ X cana-1219 73 3 π‘š+4 π‘š+4 NUM cana-1219 73 4 2 2 NUM cana-1219 73 5 βŒ‰ βŒ‰ NOUN cana-1219 73 6 , , PUNCT cana-1219 73 7 say say VERB cana-1219 73 8 ⌈ ⌈ SYM cana-1219 73 9 π‘š+4 π‘š+4 NOUN cana-1219 73 10 2 2 NUM cana-1219 73 11 βŒ‰ βŒ‰ PRON cana-1219 73 12 βˆ’ βˆ’ PROPN cana-1219 73 13 1 1 NUM cana-1219 73 14 ; ; PUNCT cana-1219 73 15 π‘§βˆ—[𝑀(π‘Œπ‘š π‘§βˆ—[𝑀(π‘Œπ‘š PROPN cana-1219 73 16 ) ) PUNCT cana-1219 73 17 ] ] PUNCT cana-1219 73 18 exists exist VERB cana-1219 73 19 ( ( PUNCT cana-1219 73 20 π‘š π‘š PROPN cana-1219 73 21 2 2 NUM cana-1219 73 22 ) ) PUNCT cana-1219 73 23 βˆ’ βˆ’ PROPN cana-1219 73 24 𝐾1,3 𝐾1,3 NOUN cana-1219 73 25 stars star NOUN cana-1219 73 26 , , PUNCT cana-1219 73 27 is be AUX cana-1219 73 28 colored color VERB cana-1219 73 29 with with ADP cana-1219 73 30 π‘š π‘š PROPN cana-1219 73 31 2 2 NUM cana-1219 73 32 colors color NOUN cana-1219 73 33 . . PUNCT cana-1219 74 1 hence hence ADV cana-1219 74 2 , , PUNCT cana-1219 74 3 there there PRON cana-1219 74 4 exists exist VERB cana-1219 74 5 an an DET cana-1219 74 6 independent independent ADJ cana-1219 74 7 set set NOUN cana-1219 74 8 remains remain VERB cana-1219 74 9 uncolored uncolored ADJ cana-1219 74 10 we we PRON cana-1219 74 11 need need VERB cana-1219 74 12 two two NUM cana-1219 74 13 more more ADJ cana-1219 74 14 colors color NOUN cana-1219 74 15 to to PART cana-1219 74 16 complete complete VERB cana-1219 74 17 the the DET cana-1219 74 18 graph graph NOUN cana-1219 74 19 . . PUNCT cana-1219 75 1 therefore therefore ADV cana-1219 75 2 , , PUNCT cana-1219 75 3 assumption assumption NOUN cana-1219 75 4 is be AUX cana-1219 75 5 contradictory contradictory ADJ cana-1219 75 6 . . PUNCT cana-1219 76 1 hence hence ADV cana-1219 76 2 π‘§βˆ—[𝑀(π‘Œπ‘š π‘§βˆ—[𝑀(π‘Œπ‘š PROPN cana-1219 76 3 ) ) PUNCT cana-1219 76 4 ] ] PUNCT cana-1219 77 1 = = PUNCT cana-1219 77 2 ⌈ ⌈ SYM cana-1219 77 3 π‘š+4 π‘š+4 NOUN cana-1219 77 4 2 2 NUM cana-1219 77 5 βŒ‰. βŒ‰. ADV cana-1219 77 6 4.1 4.1 NUM cana-1219 77 7 : : PUNCT cana-1219 77 8 star star NOUN cana-1219 77 9 cochromatic cochromatic ADJ cana-1219 77 10 number number NOUN cana-1219 77 11 of of ADP cana-1219 77 12 sunlet sunlet NOUN cana-1219 77 13 graphs graph NOUN cana-1219 77 14 theorem theorem VERB cana-1219 77 15 4.1.1 4.1.1 NUM cana-1219 77 16 : : PUNCT cana-1219 77 17 the the DET cana-1219 77 18 star star ADJ cana-1219 77 19 chromatic chromatic ADJ cana-1219 77 20 number number NOUN cana-1219 77 21 of of ADP cana-1219 77 22 sunlet sunlet NOUN cana-1219 77 23 graph graph NOUN cana-1219 77 24 π‘ž[π‘†π‘š π‘ž[π‘†π‘š PROPN cana-1219 77 25 ] ] X cana-1219 77 26 for for ADP cana-1219 77 27 π‘š π‘š PROPN cana-1219 77 28 β‰₯ β‰₯ NUM cana-1219 77 29 3 3 NUM cana-1219 77 30 is be AUX cana-1219 77 31 π‘ž[π‘†π‘š π‘ž[π‘†π‘š X cana-1219 77 32 ] ] X cana-1219 77 33 = = SYM cana-1219 77 34 ⌈ ⌈ PROPN cana-1219 77 35 π‘š+3 π‘š+3 ADP cana-1219 77 36 3 3 NUM cana-1219 77 37 βŒ‰ βŒ‰ DET cana-1219 77 38 proof proof NOUN cana-1219 77 39 : : PUNCT cana-1219 77 40 let let VERB cana-1219 77 41 π‘†π‘š π‘†π‘š PRON cana-1219 77 42 be be AUX cana-1219 77 43 the the DET cana-1219 77 44 sunlet sunlet NOUN cana-1219 77 45 graph graph NOUN cana-1219 77 46 with with ADP cana-1219 77 47 2 2 NUM cana-1219 77 48 m m NOUN cana-1219 77 49 vertices vertex NOUN cana-1219 77 50 and and CCONJ cana-1219 77 51 3 3 NUM cana-1219 77 52 m m NOUN cana-1219 77 53 edges edge NOUN cana-1219 77 54 and and CCONJ cana-1219 77 55 𝑉[π‘†π‘š 𝑉[π‘†π‘š X cana-1219 77 56 ] ] X cana-1219 77 57 = = SYM cana-1219 77 58 { { PUNCT cana-1219 77 59 𝑣𝑛 𝑣𝑛 NOUN cana-1219 77 60 : : SYM cana-1219 77 61 1 1 NUM cana-1219 77 62 ≀ ≀ NUM cana-1219 77 63 𝑛 𝑛 PRON cana-1219 77 64 ≀ ≀ NUM cana-1219 77 65 π‘š π‘š NOUN cana-1219 77 66 } } PUNCT cana-1219 77 67 βˆͺ βˆͺ ADJ cana-1219 77 68 { { PUNCT cana-1219 77 69 𝑒𝑛 𝑒𝑛 NOUN cana-1219 77 70 : : SYM cana-1219 77 71 1 1 NUM cana-1219 77 72 ≀ ≀ NUM cana-1219 77 73 𝑛 𝑛 DET cana-1219 77 74 ≀ ≀ NUM cana-1219 77 75 π‘š π‘š NOUN cana-1219 77 76 } } PUNCT cana-1219 77 77 . . PUNCT cana-1219 78 1 consider consider VERB cana-1219 78 2 the the DET cana-1219 78 3 color color NOUN cana-1219 78 4 class class NOUN cana-1219 78 5 𝐢 𝐢 PROPN cana-1219 78 6 = = PRON cana-1219 78 7 { { PUNCT cana-1219 78 8 𝑐1 𝑐1 NOUN cana-1219 78 9 , , PUNCT cana-1219 78 10 𝑐2 𝑐2 NOUN cana-1219 78 11 , , PUNCT cana-1219 78 12 𝑐3 𝑐3 NOUN cana-1219 78 13 , , PUNCT cana-1219 78 14 … … PUNCT cana-1219 78 15 , , PUNCT cana-1219 78 16 𝑐 𝑐 NOUN cana-1219 78 17 ⌈ ⌈ PROPN cana-1219 78 18 π‘š+3 π‘š+3 ADP cana-1219 78 19 3 3 NUM cana-1219 78 20 βŒ‰ βŒ‰ NOUN cana-1219 78 21 } } PUNCT cana-1219 78 22 . . PUNCT cana-1219 79 1 define define VERB cana-1219 79 2 mapping mapping NOUN cana-1219 79 3 𝜎 𝜎 NOUN cana-1219 79 4 : : PUNCT cana-1219 79 5 { { PUNCT cana-1219 79 6 𝑣𝑛 𝑣𝑛 NOUN cana-1219 79 7 : : SYM cana-1219 79 8 1 1 NUM cana-1219 79 9 ≀ ≀ NUM cana-1219 79 10 𝑛 𝑛 PRON cana-1219 79 11 ≀ ≀ NUM cana-1219 79 12 π‘š π‘š NOUN cana-1219 79 13 } } PUNCT cana-1219 79 14 βˆͺ βˆͺ ADJ cana-1219 79 15 { { PUNCT cana-1219 79 16 𝑒𝑛 𝑒𝑛 NOUN cana-1219 79 17 : : SYM cana-1219 79 18 1 1 NUM cana-1219 79 19 ≀ ≀ NUM cana-1219 79 20 𝑛 𝑛 DET cana-1219 79 21 ≀ ≀ NUM cana-1219 79 22 π‘š π‘š ADP cana-1219 79 23 } } PUNCT cana-1219 79 24 β†’ β†’ SYM cana-1219 79 25 π‘π‘˜ π‘π‘˜ NOUN cana-1219 79 26 βˆ€ βˆ€ X cana-1219 79 27 π‘˜ π‘˜ X cana-1219 79 28 = = NOUN cana-1219 79 29 1,2,3 1,2,3 NUM cana-1219 79 30 , , PUNCT cana-1219 79 31 … … PUNCT cana-1219 79 32 communications communication NOUN cana-1219 79 33 on on ADP cana-1219 79 34 applied apply VERB cana-1219 79 35 nonlinear nonlinear ADJ cana-1219 79 36 analysis analysis NOUN cana-1219 79 37 issn issn NOUN cana-1219 79 38 : : PUNCT cana-1219 79 39 1074 1074 NUM cana-1219 79 40 - - PUNCT cana-1219 79 41 133x 133x NUM cana-1219 79 42 vol vol NOUN cana-1219 79 43 31 31 NUM cana-1219 79 44 no no NOUN cana-1219 79 45 . . PUNCT cana-1219 80 1 6s 6s NUM cana-1219 80 2 ( ( PUNCT cana-1219 80 3 2024 2024 NUM cana-1219 80 4 ) ) PUNCT cana-1219 80 5 256 256 NUM cana-1219 80 6 https://internationalpubls.com https://internationalpubls.com X cana-1219 80 7 case-(i case-(i PROPN cana-1219 80 8 ): ): PUNCT cana-1219 80 9 π’Ž π’Ž PROPN cana-1219 80 10 ≑ ≑ PROPN cana-1219 80 11 𝟎 𝟎 PROPN cana-1219 80 12 π’Žπ’π’… π’Žπ’π’… VERB cana-1219 80 13 πŸ‘ πŸ‘ NUM cana-1219 80 14 β€’ β€’ NOUN cana-1219 80 15 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 PROPN cana-1219 80 16 , , PUNCT cana-1219 80 17 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 80 18 , , PUNCT cana-1219 80 19 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 80 20 , , PUNCT cana-1219 80 21 𝑒6π‘˜βˆ’4 𝑒6π‘˜βˆ’4 PROPN cana-1219 80 22 ) ) PUNCT cana-1219 80 23 = = SYM cana-1219 81 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 81 2 β€’ β€’ NUM cana-1219 81 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 81 4 , , PUNCT cana-1219 81 5 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 81 6 , , PUNCT cana-1219 81 7 𝑣6π‘˜ 𝑣6π‘˜ X cana-1219 81 8 , , PUNCT cana-1219 81 9 𝑒6π‘˜βˆ’1 𝑒6π‘˜βˆ’1 PROPN cana-1219 81 10 ) ) PUNCT cana-1219 81 11 = = PUNCT cana-1219 82 1 𝑐2π‘˜ 𝑐2π‘˜ NOUN cana-1219 82 2 β€’ β€’ NUM cana-1219 82 3 𝜎(𝑒3π‘˜βˆ’2 𝜎(𝑒3π‘˜βˆ’2 PROPN cana-1219 82 4 , , PUNCT cana-1219 82 5 𝑒3π‘˜ 𝑒3π‘˜ X cana-1219 82 6 ) ) PUNCT cana-1219 82 7 = = SYM cana-1219 82 8 𝑐 𝑐 PROPN cana-1219 82 9 ⌈ ⌈ PROPN cana-1219 82 10 π‘š+3 π‘š+3 ADP cana-1219 82 11 3 3 NUM cana-1219 82 12 βŒ‰ βŒ‰ PRON cana-1219 82 13 to to PART cana-1219 82 14 prove prove VERB cana-1219 82 15 : : PUNCT cana-1219 82 16 π‘§βˆ—[π‘†π‘š π‘§βˆ—[π‘†π‘š ADJ cana-1219 82 17 ] ] PUNCT cana-1219 82 18 < < X cana-1219 82 19 ⌈ ⌈ X cana-1219 82 20 π‘š+3 π‘š+3 ADP cana-1219 82 21 3 3 NUM cana-1219 82 22 βŒ‰ βŒ‰ NOUN cana-1219 82 23 ; ; PUNCT cana-1219 82 24 π‘§βˆ—[π‘†π‘š π‘§βˆ—[π‘†π‘š NOUN cana-1219 82 25 ] ] PUNCT cana-1219 82 26 exists exist VERB cana-1219 82 27 3π‘š 3π‘š NUM cana-1219 82 28 βˆ’ βˆ’ PROPN cana-1219 82 29 𝐾1,3 𝐾1,3 NOUN cana-1219 82 30 stars star NOUN cana-1219 82 31 , , PUNCT cana-1219 82 32 each each DET cana-1219 82 33 color color NOUN cana-1219 82 34 class class NOUN cana-1219 82 35 is be AUX cana-1219 82 36 colored color VERB cana-1219 82 37 with with ADP cana-1219 82 38 one one NUM cana-1219 82 39 color color NOUN cana-1219 82 40 . . PUNCT cana-1219 83 1 there there PRON cana-1219 83 2 exists exist VERB cana-1219 83 3 an an DET cana-1219 83 4 independent independent ADJ cana-1219 83 5 set set NOUN cana-1219 83 6 ( ( PUNCT cana-1219 83 7 𝑒3π‘˜βˆ’2 𝑒3π‘˜βˆ’2 NOUN cana-1219 83 8 , , PUNCT cana-1219 83 9 𝑒3π‘˜ 𝑒3π‘˜ X cana-1219 83 10 ) ) PUNCT cana-1219 83 11 remains remain VERB cana-1219 83 12 uncolored uncolored ADJ cana-1219 83 13 we we PRON cana-1219 83 14 need need VERB cana-1219 83 15 one one NUM cana-1219 83 16 more more ADJ cana-1219 83 17 color color NOUN cana-1219 83 18 to to PART cana-1219 83 19 complete complete VERB cana-1219 83 20 the the DET cana-1219 83 21 graph graph NOUN cana-1219 83 22 . . PUNCT cana-1219 84 1 therefore therefore ADV cana-1219 84 2 , , PUNCT cana-1219 84 3 assumption assumption NOUN cana-1219 84 4 is be AUX cana-1219 84 5 contradictory contradictory ADJ cana-1219 84 6 . . PUNCT cana-1219 85 1 hence hence ADV cana-1219 85 2 π‘ž[π‘†π‘š π‘ž[π‘†π‘š X cana-1219 85 3 ] ] PUNCT cana-1219 85 4 = = PUNCT cana-1219 85 5 ⌈ ⌈ PROPN cana-1219 85 6 π‘š+3 π‘š+3 ADP cana-1219 85 7 3 3 NUM cana-1219 85 8 βŒ‰ βŒ‰ NOUN cana-1219 85 9 . . PUNCT cana-1219 86 1 case-(ii case-(ii VERB cana-1219 86 2 ): ): PUNCT cana-1219 86 3 π’Ž π’Ž PROPN cana-1219 86 4 ≑ ≑ PROPN cana-1219 86 5 𝟏 𝟏 NUM cana-1219 86 6 π’Žπ’π’… π’Žπ’π’… VERB cana-1219 86 7 πŸ‘ πŸ‘ NUM cana-1219 86 8 β€’ β€’ NUM cana-1219 86 9 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 PROPN cana-1219 86 10 , , PUNCT cana-1219 86 11 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 86 12 , , PUNCT cana-1219 86 13 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 86 14 , , PUNCT cana-1219 86 15 𝑒6π‘˜βˆ’4 𝑒6π‘˜βˆ’4 PROPN cana-1219 86 16 ) ) PUNCT cana-1219 86 17 = = SYM cana-1219 87 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 87 2 β€’ β€’ NUM cana-1219 87 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 87 4 , , PUNCT cana-1219 87 5 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 87 6 , , PUNCT cana-1219 87 7 𝑣6π‘˜ 𝑣6π‘˜ X cana-1219 87 8 , , PUNCT cana-1219 87 9 𝑒6π‘˜βˆ’1 𝑒6π‘˜βˆ’1 PROPN cana-1219 87 10 ) ) PUNCT cana-1219 87 11 = = PUNCT cana-1219 87 12 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 87 13 β€’ β€’ NUM cana-1219 87 14 𝜎(π‘£π‘š 𝜎(π‘£π‘š PROPN cana-1219 87 15 , , PUNCT cana-1219 87 16 π‘’π‘š π‘’π‘š NOUN cana-1219 87 17 ) ) PUNCT cana-1219 87 18 = = SYM cana-1219 87 19 𝑐 𝑐 PROPN cana-1219 87 20 ⌈ ⌈ NUM cana-1219 87 21 π‘š+3 π‘š+3 ADP cana-1219 87 22 3 3 NUM cana-1219 87 23 βŒ‰βˆ’1 βŒ‰βˆ’1 PROPN cana-1219 87 24 β€’ β€’ NUM cana-1219 87 25 𝜎(𝑒3π‘˜βˆ’2 𝜎(𝑒3π‘˜βˆ’2 PROPN cana-1219 87 26 , , PUNCT cana-1219 87 27 𝑒3π‘˜ 𝑒3π‘˜ X cana-1219 87 28 ) ) PUNCT cana-1219 87 29 = = SYM cana-1219 87 30 𝑐 𝑐 PROPN cana-1219 87 31 ⌈ ⌈ NUM cana-1219 87 32 π‘š+3 π‘š+3 ADP cana-1219 87 33 3 3 NUM cana-1219 87 34 βŒ‰ βŒ‰ X cana-1219 87 35 case-(iii case-(iii NOUN cana-1219 87 36 ): ): PUNCT cana-1219 87 37 π’Ž π’Ž PROPN cana-1219 87 38 ≑ ≑ PROPN cana-1219 87 39 𝟐 𝟐 NUM cana-1219 87 40 π’Žπ’π’… π’Žπ’π’… NOUN cana-1219 87 41 πŸ‘ πŸ‘ NUM cana-1219 87 42 β€’ β€’ NOUN cana-1219 87 43 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 PROPN cana-1219 87 44 , , PUNCT cana-1219 87 45 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 87 46 , , PUNCT cana-1219 87 47 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 87 48 , , PUNCT cana-1219 87 49 𝑒6π‘˜βˆ’4 𝑒6π‘˜βˆ’4 PROPN cana-1219 87 50 ) ) PUNCT cana-1219 87 51 = = SYM cana-1219 88 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 88 2 β€’ β€’ NUM cana-1219 88 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 88 4 , , PUNCT cana-1219 88 5 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 88 6 , , PUNCT cana-1219 88 7 𝑣6π‘˜ 𝑣6π‘˜ X cana-1219 88 8 , , PUNCT cana-1219 88 9 𝑒6π‘˜βˆ’1 𝑒6π‘˜βˆ’1 PROPN cana-1219 88 10 ) ) PUNCT cana-1219 88 11 = = PUNCT cana-1219 88 12 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 88 13 β€’ β€’ NUM cana-1219 88 14 𝜎(π‘£π‘š 𝜎(π‘£π‘š PROPN cana-1219 88 15 , , PUNCT cana-1219 88 16 π‘£π‘šβˆ’1 π‘£π‘šβˆ’1 ADJ cana-1219 88 17 ) ) PUNCT cana-1219 88 18 = = SYM cana-1219 88 19 𝑐 𝑐 PROPN cana-1219 88 20 ⌈ ⌈ NUM cana-1219 88 21 π‘š+3 π‘š+3 ADP cana-1219 88 22 3 3 NUM cana-1219 88 23 βŒ‰βˆ’1 βŒ‰βˆ’1 PROPN cana-1219 88 24 β€’ β€’ NUM cana-1219 88 25 𝜎(𝑒3π‘˜βˆ’2 𝜎(𝑒3π‘˜βˆ’2 PROPN cana-1219 88 26 , , PUNCT cana-1219 88 27 𝑒3π‘˜ 𝑒3π‘˜ X cana-1219 88 28 , , PUNCT cana-1219 88 29 π‘’π‘š π‘’π‘š NOUN cana-1219 88 30 ) ) PUNCT cana-1219 88 31 = = SYM cana-1219 88 32 𝑐 𝑐 PROPN cana-1219 88 33 ⌈ ⌈ PROPN cana-1219 88 34 π‘š+3 π‘š+3 ADP cana-1219 88 35 3 3 NUM cana-1219 88 36 βŒ‰ βŒ‰ PRON cana-1219 88 37 to to PART cana-1219 88 38 prove prove VERB cana-1219 88 39 : : PUNCT cana-1219 88 40 π‘§βˆ—[π‘†π‘š π‘§βˆ—[π‘†π‘š ADJ cana-1219 88 41 ] ] PUNCT cana-1219 88 42 < < X cana-1219 88 43 ⌈ ⌈ X cana-1219 88 44 π‘š+3 π‘š+3 ADP cana-1219 88 45 3 3 NUM cana-1219 88 46 βŒ‰ βŒ‰ NOUN cana-1219 88 47 ; ; PUNCT cana-1219 88 48 π‘§βˆ—[π‘†π‘š π‘§βˆ—[π‘†π‘š NOUN cana-1219 88 49 ] ] PUNCT cana-1219 88 50 exists exist VERB cana-1219 88 51 3π‘š 3π‘š NUM cana-1219 88 52 βˆ’ βˆ’ PROPN cana-1219 88 53 𝐾1,3 𝐾1,3 NOUN cana-1219 88 54 stars star NOUN cana-1219 88 55 , , PUNCT cana-1219 88 56 each each DET cana-1219 88 57 color color NOUN cana-1219 88 58 class class NOUN cana-1219 88 59 is be AUX cana-1219 88 60 colored color VERB cana-1219 88 61 with with ADP cana-1219 88 62 one one NUM cana-1219 88 63 color color NOUN cana-1219 88 64 . . PUNCT cana-1219 89 1 hence hence ADV cana-1219 89 2 , , PUNCT cana-1219 89 3 there there PRON cana-1219 89 4 exists exist VERB cana-1219 89 5 an an DET cana-1219 89 6 independent independent ADJ cana-1219 89 7 set set NOUN cana-1219 89 8 and and CCONJ cana-1219 89 9 clique clique NOUN cana-1219 89 10 remains remain VERB cana-1219 89 11 uncolored uncolored ADJ cana-1219 89 12 we we PRON cana-1219 89 13 need need VERB cana-1219 89 14 two two NUM cana-1219 89 15 more more ADJ cana-1219 89 16 colors color NOUN cana-1219 89 17 to to PART cana-1219 89 18 complete complete VERB cana-1219 89 19 the the DET cana-1219 89 20 graph graph NOUN cana-1219 89 21 . . PUNCT cana-1219 90 1 assumption assumption NOUN cana-1219 90 2 is be AUX cana-1219 90 3 contradictory contradictory ADJ cana-1219 90 4 . . PUNCT cana-1219 91 1 hence hence ADV cana-1219 91 2 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š ADP cana-1219 91 3 ] ] X cana-1219 91 4 = = SYM cana-1219 91 5 ⌈ ⌈ PROPN cana-1219 91 6 π‘š+3 π‘š+3 ADP cana-1219 91 7 3 3 NUM cana-1219 91 8 βŒ‰ βŒ‰ NOUN cana-1219 91 9 . . PUNCT cana-1219 92 1 4.2 4.2 NUM cana-1219 92 2 : : PUNCT cana-1219 92 3 star star NOUN cana-1219 92 4 cochromatic cochromatic ADJ cana-1219 92 5 number number NOUN cana-1219 92 6 of of ADP cana-1219 92 7 line line NOUN cana-1219 92 8 graph graph NOUN cana-1219 92 9 of of ADP cana-1219 92 10 sunlet sunlet NOUN cana-1219 92 11 graphs graph NOUN cana-1219 92 12 theorem theorem VERB cana-1219 92 13 4.2.1 4.2.1 NUM cana-1219 92 14 : : PUNCT cana-1219 92 15 the the DET cana-1219 92 16 star star ADJ cana-1219 92 17 chromatic chromatic ADJ cana-1219 92 18 number number NOUN cana-1219 92 19 of of ADP cana-1219 92 20 line line NOUN cana-1219 92 21 graph graph NOUN cana-1219 92 22 of of ADP cana-1219 92 23 sunlet sunlet NOUN cana-1219 92 24 graph graph NOUN cana-1219 92 25 π‘ž[𝐿(π‘†π‘š π‘ž[𝐿(π‘†π‘š PROPN cana-1219 92 26 ) ) PUNCT cana-1219 92 27 ] ] PUNCT cana-1219 92 28 for for ADP cana-1219 92 29 π‘š π‘š PROPN cana-1219 92 30 β‰₯ β‰₯ NUM cana-1219 92 31 4 4 NUM cana-1219 92 32 is be AUX cana-1219 92 33 π‘ž[π‘Œπ‘š π‘ž[π‘Œπ‘š SYM cana-1219 92 34 ] ] X cana-1219 92 35 = = SYM cana-1219 92 36 ⌈ ⌈ NOUN cana-1219 92 37 π‘š π‘š ADP cana-1219 92 38 2 2 NUM cana-1219 92 39 βŒ‰ βŒ‰ NOUN cana-1219 92 40 proof proof NOUN cana-1219 92 41 : : PUNCT cana-1219 92 42 let let VERB cana-1219 92 43 π‘†π‘š π‘†π‘š PRON cana-1219 92 44 be be AUX cana-1219 92 45 sunlet sunlet NOUN cana-1219 92 46 graph graph NOUN cana-1219 92 47 with with ADP cana-1219 92 48 2 2 NUM cana-1219 92 49 m m NOUN cana-1219 92 50 vertices vertex NOUN cana-1219 92 51 and and CCONJ cana-1219 92 52 3 3 NUM cana-1219 92 53 m m NOUN cana-1219 92 54 edges edge NOUN cana-1219 92 55 and and CCONJ cana-1219 92 56 𝑉[π‘†π‘š 𝑉[π‘†π‘š X cana-1219 92 57 ] ] X cana-1219 92 58 = = SYM cana-1219 92 59 { { PUNCT cana-1219 92 60 𝑣𝑛 𝑣𝑛 NOUN cana-1219 92 61 : : SYM cana-1219 92 62 1 1 NUM cana-1219 92 63 ≀ ≀ NUM cana-1219 92 64 𝑛 𝑛 DET cana-1219 92 65 ≀ ≀ NUM cana-1219 92 66 π‘š π‘š NOUN cana-1219 92 67 } } PUNCT cana-1219 92 68 βˆͺ βˆͺ ADJ cana-1219 92 69 { { PUNCT cana-1219 92 70 𝑒𝑛 𝑒𝑛 NOUN cana-1219 92 71 : : SYM cana-1219 92 72 1 1 NUM cana-1219 92 73 ≀ ≀ NUM cana-1219 92 74 𝑛 𝑛 DET cana-1219 92 75 ≀ ≀ NUM cana-1219 92 76 π‘š π‘š NOUN cana-1219 92 77 } } PUNCT cana-1219 92 78 and and CCONJ cana-1219 92 79 [ [ X cana-1219 92 80 𝐿(π‘†π‘š 𝐿(π‘†π‘š NUM cana-1219 92 81 ) ) PUNCT cana-1219 92 82 ] ] PUNCT cana-1219 92 83 be be AUX cana-1219 92 84 line line NOUN cana-1219 92 85 graph graph NOUN cana-1219 92 86 of of ADP cana-1219 92 87 prism prism NOUN cana-1219 92 88 graph graph NOUN cana-1219 92 89 with with ADP cana-1219 92 90 3 3 NUM cana-1219 92 91 m m NOUN cana-1219 92 92 vertices vertex NOUN cana-1219 92 93 where where SCONJ cana-1219 92 94 𝑉[𝐿(π‘†π‘š 𝑉[𝐿(π‘†π‘š VERB cana-1219 92 95 ) ) PUNCT cana-1219 92 96 ] ] PUNCT cana-1219 93 1 = = PUNCT cana-1219 93 2 { { PUNCT cana-1219 93 3 𝑣𝑛 𝑣𝑛 NOUN cana-1219 93 4 : : SYM cana-1219 93 5 1 1 NUM cana-1219 93 6 ≀ ≀ NUM cana-1219 93 7 𝑛 𝑛 PRON cana-1219 93 8 ≀ ≀ NUM cana-1219 93 9 π‘š π‘š NOUN cana-1219 93 10 } } PUNCT cana-1219 93 11 βˆͺ βˆͺ ADJ cana-1219 93 12 { { PUNCT cana-1219 93 13 𝑒𝑛 𝑒𝑛 NOUN cana-1219 93 14 : : SYM cana-1219 93 15 1 1 NUM cana-1219 93 16 ≀ ≀ NUM cana-1219 93 17 𝑛 𝑛 DET cana-1219 93 18 ≀ ≀ NUM cana-1219 93 19 π‘š π‘š NOUN cana-1219 93 20 } } PUNCT cana-1219 93 21 . . PUNCT cana-1219 94 1 consider consider VERB cana-1219 94 2 the the DET cana-1219 94 3 color color NOUN cana-1219 94 4 class class NOUN cana-1219 94 5 𝐢 𝐢 PROPN cana-1219 94 6 = = PRON cana-1219 94 7 { { PUNCT cana-1219 94 8 𝑐1 𝑐1 NOUN cana-1219 94 9 , , PUNCT cana-1219 94 10 𝑐2 𝑐2 NOUN cana-1219 94 11 , , PUNCT cana-1219 94 12 𝑐3 𝑐3 NOUN cana-1219 94 13 , , PUNCT cana-1219 94 14 … … PUNCT cana-1219 94 15 , , PUNCT cana-1219 94 16 𝑐 𝑐 PROPN cana-1219 94 17 ⌈ ⌈ NOUN cana-1219 94 18 π‘š π‘š ADP cana-1219 94 19 2 2 NUM cana-1219 94 20 βŒ‰ βŒ‰ NOUN cana-1219 94 21 } } PUNCT cana-1219 94 22 . . PUNCT cana-1219 95 1 define define VERB cana-1219 95 2 mapping mapping NOUN cana-1219 95 3 𝜎 𝜎 NOUN cana-1219 95 4 : : PUNCT cana-1219 95 5 { { PUNCT cana-1219 95 6 𝑣𝑛 𝑣𝑛 NOUN cana-1219 95 7 : : SYM cana-1219 95 8 1 1 NUM cana-1219 95 9 ≀ ≀ NUM cana-1219 95 10 𝑛 𝑛 PRON cana-1219 95 11 ≀ ≀ NUM cana-1219 95 12 π‘š π‘š NOUN cana-1219 95 13 } } PUNCT cana-1219 95 14 βˆͺ βˆͺ ADJ cana-1219 95 15 { { PUNCT cana-1219 95 16 𝑒𝑛 𝑒𝑛 NOUN cana-1219 95 17 : : SYM cana-1219 95 18 1 1 NUM cana-1219 95 19 ≀ ≀ NUM cana-1219 95 20 𝑛 𝑛 DET cana-1219 95 21 ≀ ≀ NUM cana-1219 95 22 π‘š π‘š ADP cana-1219 95 23 } } PUNCT cana-1219 95 24 β†’ β†’ SYM cana-1219 95 25 π‘π‘˜ π‘π‘˜ NOUN cana-1219 95 26 βˆ€ βˆ€ X cana-1219 95 27 π‘˜ π‘˜ X cana-1219 95 28 = = NOUN cana-1219 95 29 1,2,3 1,2,3 NUM cana-1219 95 30 , , PUNCT cana-1219 95 31 … … PUNCT cana-1219 95 32 communications communication NOUN cana-1219 95 33 on on ADP cana-1219 95 34 applied apply VERB cana-1219 95 35 nonlinear nonlinear ADJ cana-1219 95 36 analysis analysis NOUN cana-1219 95 37 issn issn NOUN cana-1219 95 38 : : PUNCT cana-1219 95 39 1074 1074 NUM cana-1219 95 40 - - PUNCT cana-1219 95 41 133x 133x NUM cana-1219 95 42 vol vol NOUN cana-1219 95 43 31 31 NUM cana-1219 95 44 no no NOUN cana-1219 95 45 . . PUNCT cana-1219 96 1 6s 6s NUM cana-1219 96 2 ( ( PUNCT cana-1219 96 3 2024 2024 NUM cana-1219 96 4 ) ) PUNCT cana-1219 96 5 257 257 NUM cana-1219 97 1 https://internationalpubls.com https://internationalpubls.com X cana-1219 97 2 case-(i case-(i PROPN cana-1219 97 3 ): ): PUNCT cana-1219 97 4 π’Ž π’Ž PROPN cana-1219 97 5 ≑ ≑ PROPN cana-1219 97 6 𝟎 𝟎 PROPN cana-1219 97 7 π’Žπ’π’… π’Žπ’π’… VERB cana-1219 97 8 πŸ‘ πŸ‘ NUM cana-1219 97 9 β€’ β€’ NOUN cana-1219 97 10 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 PROPN cana-1219 97 11 , , PUNCT cana-1219 97 12 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 97 13 , , PUNCT cana-1219 97 14 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 97 15 , , PUNCT cana-1219 97 16 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 97 17 , , PUNCT cana-1219 97 18 𝑒6π‘˜βˆ’4 𝑒6π‘˜βˆ’4 PROPN cana-1219 97 19 ) ) PUNCT cana-1219 97 20 = = SYM cana-1219 98 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 98 2 β€’ β€’ NUM cana-1219 98 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 98 4 , , PUNCT cana-1219 98 5 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 98 6 , , PUNCT cana-1219 98 7 𝑣6π‘˜ 𝑣6π‘˜ PROPN cana-1219 98 8 , , PUNCT cana-1219 98 9 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 98 10 , , PUNCT cana-1219 98 11 𝑒6π‘˜βˆ’1 𝑒6π‘˜βˆ’1 PROPN cana-1219 98 12 ) ) PUNCT cana-1219 98 13 = = PUNCT cana-1219 98 14 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 98 15 β€’ β€’ NOUN cana-1219 98 16 𝜎(𝑒3π‘˜ 𝜎(𝑒3π‘˜ X cana-1219 98 17 ) ) PUNCT cana-1219 98 18 = = SYM cana-1219 98 19 𝑐 𝑐 PROPN cana-1219 98 20 ⌈ ⌈ X cana-1219 98 21 π‘š π‘š ADP cana-1219 98 22 2 2 NUM cana-1219 98 23 βŒ‰ βŒ‰ X cana-1219 98 24 case-(ii case-(ii PROPN cana-1219 98 25 ): ): PUNCT cana-1219 98 26 π’Ž π’Ž PROPN cana-1219 98 27 ≑ ≑ PROPN cana-1219 98 28 𝟏 𝟏 NUM cana-1219 98 29 π’Žπ’π’… π’Žπ’π’… VERB cana-1219 98 30 πŸ‘ πŸ‘ NUM cana-1219 98 31 β€’ β€’ NUM cana-1219 98 32 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 PROPN cana-1219 98 33 , , PUNCT cana-1219 98 34 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 98 35 , , PUNCT cana-1219 98 36 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 98 37 , , PUNCT cana-1219 98 38 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 98 39 , , PUNCT cana-1219 98 40 𝑒6π‘˜βˆ’4 𝑒6π‘˜βˆ’4 PROPN cana-1219 98 41 ) ) PUNCT cana-1219 98 42 = = SYM cana-1219 99 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 99 2 β€’ β€’ NUM cana-1219 99 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 99 4 , , PUNCT cana-1219 99 5 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 99 6 , , PUNCT cana-1219 99 7 𝑣6π‘˜ 𝑣6π‘˜ PROPN cana-1219 99 8 , , PUNCT cana-1219 99 9 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 99 10 , , PUNCT cana-1219 99 11 𝑒6π‘˜βˆ’1 𝑒6π‘˜βˆ’1 PROPN cana-1219 99 12 ) ) PUNCT cana-1219 99 13 = = PUNCT cana-1219 99 14 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 99 15 β€’ β€’ NUM cana-1219 99 16 𝜎(π‘£π‘š 𝜎(π‘£π‘š PROPN cana-1219 99 17 , , PUNCT cana-1219 99 18 π‘’π‘š π‘’π‘š NOUN cana-1219 99 19 ) ) PUNCT cana-1219 99 20 = = SYM cana-1219 99 21 𝑐 𝑐 PROPN cana-1219 99 22 ⌈ ⌈ X cana-1219 99 23 π‘š π‘š ADP cana-1219 99 24 2 2 NUM cana-1219 99 25 βŒ‰βˆ’1 βŒ‰βˆ’1 PROPN cana-1219 99 26 β€’ β€’ NUM cana-1219 99 27 𝜎(𝑒3π‘˜ 𝜎(𝑒3π‘˜ X cana-1219 99 28 ) ) PUNCT cana-1219 99 29 = = SYM cana-1219 99 30 𝑐 𝑐 PROPN cana-1219 99 31 ⌈ ⌈ X cana-1219 99 32 π‘š π‘š ADP cana-1219 99 33 2 2 NUM cana-1219 99 34 βŒ‰ βŒ‰ X cana-1219 99 35 case-(iii case-(iii NOUN cana-1219 99 36 ): ): PUNCT cana-1219 99 37 π’Ž π’Ž PROPN cana-1219 99 38 ≑ ≑ PROPN cana-1219 99 39 𝟐 𝟐 NUM cana-1219 99 40 π’Žπ’π’… π’Žπ’π’… NOUN cana-1219 99 41 πŸ‘ πŸ‘ NUM cana-1219 99 42 β€’ β€’ NOUN cana-1219 99 43 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 PROPN cana-1219 99 44 , , PUNCT cana-1219 99 45 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 99 46 , , PUNCT cana-1219 99 47 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 99 48 , , PUNCT cana-1219 99 49 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 PROPN cana-1219 99 50 , , PUNCT cana-1219 99 51 𝑒6π‘˜βˆ’4 𝑒6π‘˜βˆ’4 PROPN cana-1219 99 52 ) ) PUNCT cana-1219 99 53 = = SYM cana-1219 100 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 100 2 β€’ β€’ NUM cana-1219 100 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 100 4 , , PUNCT cana-1219 100 5 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 100 6 , , PUNCT cana-1219 100 7 𝑣6π‘˜ 𝑣6π‘˜ PROPN cana-1219 100 8 , , PUNCT cana-1219 100 9 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 100 10 , , PUNCT cana-1219 100 11 𝑒6π‘˜βˆ’1 𝑒6π‘˜βˆ’1 PROPN cana-1219 100 12 ) ) PUNCT cana-1219 100 13 = = PUNCT cana-1219 100 14 𝑐2π‘˜ 𝑐2π‘˜ X cana-1219 100 15 β€’ β€’ NUM cana-1219 100 16 𝜎(π‘£π‘š 𝜎(π‘£π‘š PROPN cana-1219 100 17 , , PUNCT cana-1219 100 18 π‘£π‘šβˆ’1 π‘£π‘šβˆ’1 ADJ cana-1219 100 19 ) ) PUNCT cana-1219 100 20 = = SYM cana-1219 100 21 𝑐 𝑐 PROPN cana-1219 100 22 ⌈ ⌈ X cana-1219 100 23 π‘š π‘š ADP cana-1219 100 24 2 2 NUM cana-1219 100 25 βŒ‰βˆ’1 βŒ‰βˆ’1 PROPN cana-1219 100 26 β€’ β€’ NOUN cana-1219 100 27 𝜎(𝑒3π‘˜ 𝜎(𝑒3π‘˜ X cana-1219 100 28 , , PUNCT cana-1219 100 29 π‘’π‘š π‘’π‘š INTJ cana-1219 100 30 , , PUNCT cana-1219 100 31 π‘’π‘šβˆ’1 π‘’π‘šβˆ’1 PROPN cana-1219 100 32 ) ) PUNCT cana-1219 100 33 = = SYM cana-1219 100 34 𝑐 𝑐 PROPN cana-1219 100 35 ⌈ ⌈ X cana-1219 100 36 π‘š π‘š ADP cana-1219 100 37 2 2 NUM cana-1219 100 38 βŒ‰ βŒ‰ X cana-1219 100 39 to to PART cana-1219 100 40 prove prove VERB cana-1219 100 41 : : PUNCT cana-1219 100 42 π‘§βˆ—[𝐿(π‘†π‘š π‘§βˆ—[𝐿(π‘†π‘š PROPN cana-1219 100 43 ) ) PUNCT cana-1219 100 44 ] ] PUNCT cana-1219 100 45 < < X cana-1219 100 46 ⌈ ⌈ X cana-1219 100 47 π‘š π‘š ADP cana-1219 100 48 2 2 NUM cana-1219 100 49 βŒ‰ βŒ‰ NOUN cana-1219 100 50 , , PUNCT cana-1219 100 51 say say VERB cana-1219 100 52 ⌈ ⌈ SYM cana-1219 100 53 π‘š π‘š ADP cana-1219 100 54 2 2 NUM cana-1219 100 55 βŒ‰ βŒ‰ SCONJ cana-1219 100 56 βˆ’ βˆ’ PROPN cana-1219 100 57 1 1 NUM cana-1219 100 58 ; ; PUNCT cana-1219 100 59 π‘§βˆ—[𝐿(π‘†π‘š π‘§βˆ—[𝐿(π‘†π‘š PROPN cana-1219 100 60 ) ) PUNCT cana-1219 100 61 ] ] PUNCT cana-1219 100 62 exists exist VERB cana-1219 100 63 ( ( PUNCT cana-1219 100 64 π‘š π‘š PROPN cana-1219 100 65 3 3 NUM cana-1219 100 66 ) ) PUNCT cana-1219 100 67 βˆ’ βˆ’ PROPN cana-1219 100 68 𝐾1,3 𝐾1,3 NOUN cana-1219 100 69 stars star NOUN cana-1219 100 70 , , PUNCT cana-1219 100 71 each each DET cana-1219 100 72 color color NOUN cana-1219 100 73 class class NOUN cana-1219 100 74 is be AUX cana-1219 100 75 colored color VERB cana-1219 100 76 with with ADP cana-1219 100 77 π‘š π‘š PROPN cana-1219 100 78 3 3 NUM cana-1219 100 79 colors color NOUN cana-1219 100 80 . . PUNCT cana-1219 101 1 hence hence ADV cana-1219 101 2 , , PUNCT cana-1219 101 3 there there PRON cana-1219 101 4 exists exist VERB cana-1219 101 5 an an DET cana-1219 101 6 independent independent ADJ cana-1219 101 7 set set NOUN cana-1219 101 8 remains remain VERB cana-1219 101 9 uncolored uncolored ADJ cana-1219 101 10 we we PRON cana-1219 101 11 need need VERB cana-1219 101 12 one one NUM cana-1219 101 13 more more ADJ cana-1219 101 14 colors color NOUN cana-1219 101 15 to to PART cana-1219 101 16 complete complete VERB cana-1219 101 17 the the DET cana-1219 101 18 graph graph NOUN cana-1219 101 19 . . PUNCT cana-1219 102 1 therefore therefore ADV cana-1219 102 2 , , PUNCT cana-1219 102 3 assumption assumption NOUN cana-1219 102 4 is be AUX cana-1219 102 5 contradictory contradictory ADJ cana-1219 102 6 . . PUNCT cana-1219 103 1 hence hence ADV cana-1219 103 2 π‘§βˆ—[𝐿(π‘†π‘š π‘§βˆ—[𝐿(π‘†π‘š PROPN cana-1219 103 3 ) ) PUNCT cana-1219 103 4 ] ] PUNCT cana-1219 104 1 = = PUNCT cana-1219 104 2 ⌈ ⌈ NOUN cana-1219 104 3 π‘š π‘š ADP cana-1219 104 4 2 2 NUM cana-1219 104 5 βŒ‰. βŒ‰. ADV cana-1219 104 6 4.3 4.3 NUM cana-1219 104 7 : : PUNCT cana-1219 104 8 star star NOUN cana-1219 104 9 cochromatic cochromatic ADJ cana-1219 104 10 number number NOUN cana-1219 104 11 of of ADP cana-1219 104 12 middle middle ADJ cana-1219 104 13 graph graph NOUN cana-1219 104 14 of of ADP cana-1219 104 15 sunlet sunlet NOUN cana-1219 104 16 graphs graph NOUN cana-1219 104 17 theorem theorem VERB cana-1219 104 18 4.3.1 4.3.1 NUM cana-1219 104 19 : : PUNCT cana-1219 104 20 the the DET cana-1219 104 21 star star NOUN cana-1219 104 22 chromatic chromatic ADJ cana-1219 104 23 number number NOUN cana-1219 104 24 of of ADP cana-1219 104 25 middle middle ADJ cana-1219 104 26 graph graph NOUN cana-1219 104 27 of of ADP cana-1219 104 28 sunlet sunlet NOUN cana-1219 104 29 graph graph NOUN cana-1219 104 30 π‘ž[𝑀(π‘†π‘š π‘ž[𝑀(π‘†π‘š PROPN cana-1219 104 31 ) ) PUNCT cana-1219 104 32 ] ] PUNCT cana-1219 104 33 for for ADP cana-1219 104 34 π‘š π‘š PROPN cana-1219 104 35 ≑ ≑ PROPN cana-1219 104 36 0 0 NUM cana-1219 104 37 π‘šπ‘œπ‘‘ π‘šπ‘œπ‘‘ NOUN cana-1219 104 38 3 3 NUM cana-1219 104 39 is be AUX cana-1219 104 40 π‘ž[𝑀(π‘†π‘š π‘ž[𝑀(π‘†π‘š PROPN cana-1219 104 41 ) ) PUNCT cana-1219 104 42 ] ] PUNCT cana-1219 105 1 = = PUNCT cana-1219 105 2 ⌈ ⌈ X cana-1219 105 3 π‘š+4 π‘š+4 NUM cana-1219 105 4 3 3 NUM cana-1219 105 5 βŒ‰ βŒ‰ DET cana-1219 105 6 proof proof NOUN cana-1219 105 7 : : PUNCT cana-1219 105 8 let let VERB cana-1219 105 9 π‘†π‘š π‘†π‘š PRON cana-1219 105 10 be be AUX cana-1219 105 11 sunlet sunlet NOUN cana-1219 105 12 graph graph NOUN cana-1219 105 13 with with ADP cana-1219 105 14 2 2 NUM cana-1219 105 15 m m NOUN cana-1219 105 16 vertices vertex NOUN cana-1219 105 17 and and CCONJ cana-1219 105 18 3 3 NUM cana-1219 105 19 m m NOUN cana-1219 105 20 edges edge NOUN cana-1219 105 21 and and CCONJ cana-1219 105 22 𝑉[π‘†π‘š 𝑉[π‘†π‘š X cana-1219 105 23 ] ] X cana-1219 105 24 = = SYM cana-1219 105 25 { { PUNCT cana-1219 105 26 𝑣𝑛 𝑣𝑛 NOUN cana-1219 105 27 : : SYM cana-1219 105 28 1 1 NUM cana-1219 105 29 ≀ ≀ NUM cana-1219 105 30 𝑛 𝑛 PRON cana-1219 105 31 ≀ ≀ NUM cana-1219 105 32 π‘š π‘š NOUN cana-1219 105 33 } } PUNCT cana-1219 105 34 βˆͺ βˆͺ ADJ cana-1219 105 35 { { PUNCT cana-1219 105 36 𝑒𝑛 𝑒𝑛 NOUN cana-1219 105 37 : : SYM cana-1219 105 38 1 1 NUM cana-1219 105 39 ≀ ≀ NUM cana-1219 105 40 𝑛 𝑛 DET cana-1219 105 41 ≀ ≀ NUM cana-1219 105 42 π‘š π‘š NOUN cana-1219 105 43 } } PUNCT cana-1219 105 44 and and CCONJ cana-1219 105 45 [ [ X cana-1219 105 46 𝑀(π‘†π‘š 𝑀(π‘†π‘š NOUN cana-1219 105 47 ) ) PUNCT cana-1219 105 48 ] ] PUNCT cana-1219 105 49 be be AUX cana-1219 105 50 middle middle ADJ cana-1219 105 51 graph graph NOUN cana-1219 105 52 of of ADP cana-1219 105 53 sunlet sunlet NOUN cana-1219 105 54 graph graph NOUN cana-1219 105 55 where where SCONJ cana-1219 105 56 subdividing subdivide VERB cana-1219 105 57 each each DET cana-1219 105 58 edge edge NOUN cana-1219 105 59 exactly exactly ADV cana-1219 105 60 once once ADV cana-1219 105 61 and and CCONJ cana-1219 105 62 join join VERB cana-1219 105 63 the the DET cana-1219 105 64 adjacent adjacent ADJ cana-1219 105 65 vertices vertex NOUN cana-1219 105 66 , , PUNCT cana-1219 105 67 the the DET cana-1219 105 68 vertex vertex NOUN cana-1219 105 69 set set NOUN cana-1219 105 70 of of ADP cana-1219 105 71 𝑉[𝑀(π‘†π‘š 𝑉[𝑀(π‘†π‘š PROPN cana-1219 105 72 ) ) PUNCT cana-1219 105 73 ] ] PUNCT cana-1219 106 1 = = PUNCT cana-1219 106 2 { { PUNCT cana-1219 106 3 𝑣𝑛 𝑣𝑛 NOUN cana-1219 106 4 : : SYM cana-1219 106 5 1 1 NUM cana-1219 106 6 ≀ ≀ NUM cana-1219 106 7 𝑛 𝑛 PRON cana-1219 106 8 ≀ ≀ NUM cana-1219 106 9 π‘š π‘š NOUN cana-1219 106 10 } } PUNCT cana-1219 106 11 βˆͺ βˆͺ ADJ cana-1219 106 12 { { PUNCT cana-1219 106 13 𝑣𝑛 𝑣𝑛 NOUN cana-1219 106 14 β€² β€² NUM cana-1219 106 15 : : PUNCT cana-1219 106 16 1 1 NUM cana-1219 106 17 ≀ ≀ NUM cana-1219 106 18 𝑛 𝑛 PRON cana-1219 106 19 ≀ ≀ NUM cana-1219 106 20 π‘š π‘š NOUN cana-1219 106 21 } } PUNCT cana-1219 106 22 βˆͺ βˆͺ ADJ cana-1219 106 23 { { PUNCT cana-1219 106 24 𝑒𝑛 𝑒𝑛 NOUN cana-1219 106 25 : : SYM cana-1219 106 26 1 1 NUM cana-1219 106 27 ≀ ≀ NUM cana-1219 106 28 𝑛 𝑛 PRON cana-1219 106 29 ≀ ≀ NUM cana-1219 106 30 π‘š π‘š NOUN cana-1219 106 31 } } PUNCT cana-1219 106 32 βˆͺ βˆͺ ADJ cana-1219 106 33 { { PUNCT cana-1219 106 34 𝑒𝑛 𝑒𝑛 NOUN cana-1219 106 35 β€² β€² NUM cana-1219 106 36 : : PUNCT cana-1219 106 37 1 1 NUM cana-1219 106 38 ≀ ≀ NUM cana-1219 106 39 𝑛 𝑛 DET cana-1219 106 40 ≀ ≀ NUM cana-1219 106 41 π‘š π‘š NOUN cana-1219 106 42 } } PUNCT cana-1219 106 43 . . PUNCT cana-1219 107 1 consider consider VERB cana-1219 107 2 the the DET cana-1219 107 3 color color NOUN cana-1219 107 4 class class NOUN cana-1219 107 5 𝐢 𝐢 PROPN cana-1219 107 6 = = PRON cana-1219 107 7 { { PUNCT cana-1219 107 8 𝑐1 𝑐1 NOUN cana-1219 107 9 , , PUNCT cana-1219 107 10 𝑐2 𝑐2 NOUN cana-1219 107 11 , , PUNCT cana-1219 107 12 𝑐3 𝑐3 NOUN cana-1219 107 13 , , PUNCT cana-1219 107 14 … … PUNCT cana-1219 107 15 , , PUNCT cana-1219 107 16 𝑐 𝑐 NOUN cana-1219 107 17 ⌈ ⌈ SYM cana-1219 107 18 π‘š+4 π‘š+4 NUM cana-1219 107 19 3 3 NUM cana-1219 107 20 βŒ‰ βŒ‰ NOUN cana-1219 107 21 } } PUNCT cana-1219 107 22 . . PUNCT cana-1219 108 1 define define VERB cana-1219 108 2 mapping mapping NOUN cana-1219 108 3 𝜎 𝜎 NOUN cana-1219 108 4 : : PUNCT cana-1219 108 5 { { PUNCT cana-1219 108 6 𝑣𝑛 𝑣𝑛 NOUN cana-1219 108 7 : : SYM cana-1219 108 8 1 1 NUM cana-1219 108 9 ≀ ≀ NUM cana-1219 108 10 𝑛 𝑛 PRON cana-1219 108 11 ≀ ≀ NUM cana-1219 108 12 π‘š π‘š NOUN cana-1219 108 13 } } PUNCT cana-1219 108 14 βˆͺ βˆͺ ADJ cana-1219 108 15 { { PUNCT cana-1219 108 16 𝑣𝑛 𝑣𝑛 NOUN cana-1219 108 17 β€² β€² NUM cana-1219 108 18 : : PUNCT cana-1219 108 19 1 1 NUM cana-1219 108 20 ≀ ≀ NUM cana-1219 108 21 𝑛 𝑛 PRON cana-1219 108 22 ≀ ≀ NUM cana-1219 108 23 π‘š π‘š NOUN cana-1219 108 24 } } PUNCT cana-1219 108 25 βˆͺ βˆͺ ADJ cana-1219 108 26 { { PUNCT cana-1219 108 27 𝑒𝑛 𝑒𝑛 NOUN cana-1219 108 28 : : SYM cana-1219 108 29 1 1 NUM cana-1219 108 30 ≀ ≀ NUM cana-1219 108 31 𝑛 𝑛 PRON cana-1219 108 32 ≀ ≀ NUM cana-1219 108 33 π‘š π‘š NOUN cana-1219 108 34 } } PUNCT cana-1219 108 35 βˆͺ βˆͺ ADJ cana-1219 108 36 { { PUNCT cana-1219 108 37 𝑒𝑛 𝑒𝑛 NOUN cana-1219 108 38 β€² β€² NUM cana-1219 108 39 : : PUNCT cana-1219 108 40 1 1 NUM cana-1219 108 41 ≀ ≀ NUM cana-1219 108 42 𝑛 𝑛 DET cana-1219 108 43 ≀ ≀ NUM cana-1219 108 44 π‘š π‘š ADP cana-1219 108 45 } } PUNCT cana-1219 108 46 β†’ β†’ SYM cana-1219 108 47 π‘π‘˜ π‘π‘˜ NOUN cana-1219 108 48 βˆ€ βˆ€ X cana-1219 108 49 π‘˜ π‘˜ X cana-1219 108 50 = = NOUN cana-1219 108 51 1,2,3 1,2,3 NUM cana-1219 108 52 , , PUNCT cana-1219 108 53 … … PUNCT cana-1219 108 54 assign assign VERB cana-1219 108 55 the the DET cana-1219 108 56 coloring coloring NOUN cana-1219 108 57 as as SCONJ cana-1219 108 58 follows follow VERB cana-1219 108 59 : : PUNCT cana-1219 108 60 β€’ β€’ NUM cana-1219 108 61 𝜎(𝑣6π‘˜βˆ’4 𝜎(𝑣6π‘˜βˆ’4 NOUN cana-1219 108 62 β€² β€² NOUN cana-1219 108 63 , , PUNCT cana-1219 108 64 𝑒6π‘›βˆ’4 𝑒6π‘›βˆ’4 PROPN cana-1219 108 65 β€² β€² NUM cana-1219 108 66 , , PUNCT cana-1219 109 1 𝑒6π‘›βˆ’3 𝑒6π‘›βˆ’3 NOUN cana-1219 109 2 β€² β€² NUM cana-1219 109 3 , , PUNCT cana-1219 109 4 𝑣6π‘˜βˆ’5 𝑣6π‘˜βˆ’5 NOUN cana-1219 109 5 β€² β€² NUM cana-1219 109 6 , , PUNCT cana-1219 109 7 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 109 8 β€² β€² NOUN cana-1219 109 9 , , PUNCT cana-1219 109 10 𝑣6π‘˜βˆ’4 𝑣6π‘˜βˆ’4 NOUN cana-1219 109 11 , , PUNCT cana-1219 109 12 𝑣6π‘˜βˆ’3 𝑣6π‘˜βˆ’3 PROPN cana-1219 109 13 ) ) PUNCT cana-1219 109 14 = = SYM cana-1219 110 1 𝑐2π‘˜βˆ’1 𝑐2π‘˜βˆ’1 PROPN cana-1219 110 2 β€’ β€’ NUM cana-1219 110 3 𝜎(𝑣6π‘˜βˆ’1 𝜎(𝑣6π‘˜βˆ’1 PROPN cana-1219 110 4 β€² β€² NOUN cana-1219 110 5 , , PUNCT cana-1219 110 6 𝑒6π‘›βˆ’1 𝑒6π‘›βˆ’1 PROPN cana-1219 110 7 β€² β€² NOUN cana-1219 110 8 , , PUNCT cana-1219 110 9 𝑒6𝑛 𝑒6𝑛 PROPN cana-1219 110 10 β€² β€² NUM cana-1219 110 11 , , PUNCT cana-1219 110 12 𝑣6π‘˜βˆ’2 𝑣6π‘˜βˆ’2 PROPN cana-1219 110 13 β€² β€² NUM cana-1219 110 14 , , PUNCT cana-1219 110 15 𝑣6π‘˜ 𝑣6π‘˜ NOUN cana-1219 110 16 β€² β€² NUM cana-1219 110 17 , , PUNCT cana-1219 110 18 𝑣6π‘˜βˆ’1 𝑣6π‘˜βˆ’1 PROPN cana-1219 110 19 , , PUNCT cana-1219 110 20 𝑣6π‘˜ 𝑣6π‘˜ X cana-1219 110 21 ) ) PUNCT cana-1219 110 22 = = PUNCT cana-1219 110 23 𝑐2π‘˜ 𝑐2π‘˜ PRON cana-1219 110 24 communications communication NOUN cana-1219 110 25 on on ADP cana-1219 110 26 applied apply VERB cana-1219 110 27 nonlinear nonlinear ADJ cana-1219 110 28 analysis analysis NOUN cana-1219 110 29 issn issn NOUN cana-1219 110 30 : : PUNCT cana-1219 110 31 1074 1074 NUM cana-1219 110 32 - - PUNCT cana-1219 110 33 133x 133x NUM cana-1219 110 34 vol vol NOUN cana-1219 110 35 31 31 NUM cana-1219 110 36 no no NOUN cana-1219 110 37 . . PUNCT cana-1219 111 1 6s 6s NUM cana-1219 111 2 ( ( PUNCT cana-1219 111 3 2024 2024 NUM cana-1219 111 4 ) ) PUNCT cana-1219 111 5 258 258 NUM cana-1219 111 6 https://internationalpubls.com https://internationalpubls.com X cana-1219 111 7 β€’ β€’ ADP cana-1219 111 8 𝜎(π‘’π‘˜ 𝜎(π‘’π‘˜ PROPN cana-1219 111 9 , , PUNCT cana-1219 111 10 𝑣3π‘˜βˆ’2 𝑣3π‘˜βˆ’2 NOUN cana-1219 111 11 ) ) PUNCT cana-1219 111 12 = = SYM cana-1219 112 1 𝑐 𝑐 PROPN cana-1219 112 2 ⌈ ⌈ X cana-1219 112 3 π‘š+4 π‘š+4 NUM cana-1219 112 4 3 3 NUM cana-1219 112 5 βŒ‰ βŒ‰ PRON cana-1219 112 6 to to PART cana-1219 112 7 prove prove VERB cana-1219 112 8 : : PUNCT cana-1219 112 9 π‘§βˆ—[𝑀(π‘†π‘š π‘§βˆ—[𝑀(π‘†π‘š PROPN cana-1219 112 10 ) ) PUNCT cana-1219 112 11 ] ] PUNCT cana-1219 113 1 < < X cana-1219 113 2 ⌈ ⌈ X cana-1219 113 3 π‘š+4 π‘š+4 NUM cana-1219 113 4 3 3 NUM cana-1219 113 5 βŒ‰ βŒ‰ NOUN cana-1219 113 6 , , PUNCT cana-1219 113 7 say say VERB cana-1219 113 8 ⌈ ⌈ SYM cana-1219 113 9 π‘š+4 π‘š+4 NOUN cana-1219 113 10 3 3 NUM cana-1219 113 11 βŒ‰ βŒ‰ SCONJ cana-1219 113 12 βˆ’ βˆ’ PROPN cana-1219 113 13 1 1 NUM cana-1219 113 14 ; ; PUNCT cana-1219 113 15 π‘§βˆ—[𝑀(π‘†π‘š π‘§βˆ—[𝑀(π‘†π‘š PROPN cana-1219 113 16 ) ) PUNCT cana-1219 113 17 ] ] PUNCT cana-1219 113 18 exists exist VERB cana-1219 113 19 ( ( PUNCT cana-1219 113 20 π‘š π‘š PROPN cana-1219 113 21 3 3 NUM cana-1219 113 22 ) ) PUNCT cana-1219 113 23 βˆ’ βˆ’ PROPN cana-1219 113 24 𝐾1,3 𝐾1,3 NOUN cana-1219 113 25 stars star NOUN cana-1219 113 26 , , PUNCT cana-1219 113 27 is be AUX cana-1219 113 28 colored color VERB cana-1219 113 29 with with ADP cana-1219 113 30 π‘š π‘š PROPN cana-1219 113 31 3 3 NUM cana-1219 113 32 colors color NOUN cana-1219 113 33 . . PUNCT cana-1219 114 1 hence hence ADV cana-1219 114 2 , , PUNCT cana-1219 114 3 there there PRON cana-1219 114 4 exists exist VERB cana-1219 114 5 an an DET cana-1219 114 6 independent independent ADJ cana-1219 114 7 set set NOUN cana-1219 114 8 remains remain VERB cana-1219 114 9 uncolored uncolored ADJ cana-1219 114 10 we we PRON cana-1219 114 11 need need VERB cana-1219 114 12 one one NUM cana-1219 114 13 more more ADJ cana-1219 114 14 color color NOUN cana-1219 114 15 to to PART cana-1219 114 16 complete complete VERB cana-1219 114 17 the the DET cana-1219 114 18 graph graph NOUN cana-1219 114 19 . . PUNCT cana-1219 115 1 assumption assumption NOUN cana-1219 115 2 is be AUX cana-1219 115 3 contradiction contradiction NOUN cana-1219 115 4 . . PUNCT cana-1219 116 1 hence hence ADV cana-1219 116 2 π‘§βˆ—[𝑀(π‘†π‘š π‘§βˆ—[𝑀(π‘†π‘š PROPN cana-1219 116 3 ) ) PUNCT cana-1219 116 4 ] ] PUNCT cana-1219 117 1 = = PUNCT cana-1219 117 2 ⌈ ⌈ SYM cana-1219 117 3 π‘š+4 π‘š+4 NUM cana-1219 117 4 3 3 NUM cana-1219 117 5 βŒ‰. βŒ‰. ADV cana-1219 117 6 5 5 NUM cana-1219 117 7 . . PUNCT cana-1219 117 8 conclusion conclusion NOUN cana-1219 117 9 : : PUNCT cana-1219 117 10 in in ADP cana-1219 117 11 this this DET cana-1219 117 12 article article NOUN cana-1219 117 13 , , PUNCT cana-1219 117 14 we we PRON cana-1219 117 15 examined examine VERB cana-1219 117 16 the the DET cana-1219 117 17 star star NOUN cana-1219 117 18 cochromatic cochromatic ADJ cana-1219 117 19 number number NOUN cana-1219 117 20 of of ADP cana-1219 117 21 prism prism NOUN cana-1219 117 22 graphs graph NOUN cana-1219 117 23 , , PUNCT cana-1219 117 24 the the DET cana-1219 117 25 line line NOUN cana-1219 117 26 graph graph NOUN cana-1219 117 27 of of ADP cana-1219 117 28 prism prism NOUN cana-1219 117 29 graphs graph NOUN cana-1219 117 30 , , PUNCT cana-1219 117 31 the the DET cana-1219 117 32 middle middle ADJ cana-1219 117 33 graph graph NOUN cana-1219 117 34 of of ADP cana-1219 117 35 prism prism NOUN cana-1219 117 36 graphs graph NOUN cana-1219 117 37 , , PUNCT cana-1219 117 38 the the DET cana-1219 117 39 star star NOUN cana-1219 117 40 cochromatic cochromatic PROPN cana-1219 117 41 index index NOUN cana-1219 117 42 for for ADP cana-1219 117 43 sunlet sunlet NOUN cana-1219 117 44 graphs graph NOUN cana-1219 117 45 , , PUNCT cana-1219 117 46 the the DET cana-1219 117 47 line line NOUN cana-1219 117 48 graph graph NOUN cana-1219 117 49 of of ADP cana-1219 117 50 sunlet sunlet NOUN cana-1219 117 51 graphs graph NOUN cana-1219 117 52 and and CCONJ cana-1219 117 53 the the DET cana-1219 117 54 middle middle ADJ cana-1219 117 55 graph graph NOUN cana-1219 117 56 of of ADP cana-1219 117 57 sunlet sunlet NOUN cana-1219 117 58 graphs graph NOUN cana-1219 117 59 . . PUNCT cana-1219 118 1 further further ADJ cana-1219 118 2 improvements improvement NOUN cana-1219 118 3 will will AUX cana-1219 118 4 encompass encompass VERB cana-1219 118 5 more more ADJ cana-1219 118 6 central central ADJ cana-1219 118 7 , , PUNCT cana-1219 118 8 line line NOUN cana-1219 118 9 , , PUNCT cana-1219 118 10 total total ADJ cana-1219 118 11 and and CCONJ cana-1219 118 12 middle middle ADJ cana-1219 118 13 graphs graph NOUN cana-1219 118 14 in in ADP cana-1219 118 15 various various ADJ cana-1219 118 16 graph graph NOUN cana-1219 118 17 families family NOUN cana-1219 118 18 . . PUNCT cana-1219 119 1 references reference NOUN cana-1219 119 2 [ [ X cana-1219 119 3 1 1 NUM cana-1219 119 4 ] ] PUNCT cana-1219 119 5 hassler hassler PROPN cana-1219 119 6 whitney whitney PROPN cana-1219 119 7 , , PUNCT cana-1219 119 8 congruent congruent ADJ cana-1219 119 9 graphs graph NOUN cana-1219 119 10 and and CCONJ cana-1219 119 11 the the DET cana-1219 119 12 connectivity connectivity NOUN cana-1219 119 13 of of ADP cana-1219 119 14 graphs graph NOUN cana-1219 119 15 , , PUNCT cana-1219 119 16 amer amer PROPN cana-1219 119 17 . . PUNCT cana-1219 120 1 j. j. PROPN cana-1219 120 2 math math PROPN cana-1219 120 3 . . PUNCT cana-1219 121 1 1932 1932 NUM cana-1219 121 2 . . PUNCT cana-1219 122 1 [ [ X cana-1219 122 2 2 2 NUM cana-1219 122 3 ] ] X cana-1219 122 4 kowsalya kowsalya NOUN cana-1219 122 5 . . PUNCT cana-1219 123 1 v v X cana-1219 123 2 , , PUNCT cana-1219 123 3 vernold vernold ADJ cana-1219 123 4 vivin vivin NOUN cana-1219 123 5 . . PUNCT cana-1219 124 1 j j PROPN cana-1219 124 2 and and CCONJ cana-1219 124 3 venkatachalam venkatachalam PROPN cana-1219 124 4 . . PUNCT cana-1219 125 1 m m PROPN cana-1219 125 2 , , PUNCT cana-1219 125 3 on on ADP cana-1219 125 4 star star ADJ cana-1219 125 5 chromatic chromatic ADJ cana-1219 125 6 number number NOUN cana-1219 125 7 of of ADP cana-1219 125 8 sunlet sunlet NOUN cana-1219 125 9 graph graph NOUN cana-1219 125 10 families family NOUN cana-1219 125 11 , , PUNCT cana-1219 125 12 ars ar VERB cana-1219 125 13 combinatoria combinatoria PROPN cana-1219 125 14 , , PUNCT cana-1219 125 15 vol-123 vol-123 NOUN cana-1219 125 16 ( ( PUNCT cana-1219 125 17 431 431 NUM cana-1219 125 18 - - SYM cana-1219 125 19 437 437 NUM cana-1219 125 20 ) ) PUNCT cana-1219 125 21 , , PUNCT cana-1219 125 22 2015 2015 NUM cana-1219 125 23 . . PUNCT cana-1219 126 1 [ [ X cana-1219 126 2 3 3 NUM cana-1219 126 3 ] ] X cana-1219 126 4 lesinak lesinak NOUN cana-1219 126 5 and and CCONJ cana-1219 126 6 straight straight ADJ cana-1219 126 7 , , PUNCT cana-1219 126 8 a a DET cana-1219 126 9 cochromatic cochromatic ADJ cana-1219 126 10 number number NOUN cana-1219 126 11 of of ADP cana-1219 126 12 a a DET cana-1219 126 13 graph graph NOUN cana-1219 126 14 , , PUNCT cana-1219 126 15 1977 1977 NUM cana-1219 126 16 . . PUNCT cana-1219 127 1 [ [ X cana-1219 127 2 4 4 NUM cana-1219 127 3 ] ] PUNCT cana-1219 127 4 m. m. NOUN cana-1219 127 5 poobalaranjani poobalaranjani NOUN cana-1219 127 6 , , PUNCT cana-1219 127 7 a a DET cana-1219 127 8 study study NOUN cana-1219 127 9 on on ADP cana-1219 127 10 cocoloring cocoloring NOUN cana-1219 127 11 and and CCONJ cana-1219 127 12 variants variant NOUN cana-1219 127 13 of of ADP cana-1219 127 14 cocoloring cocolore VERB cana-1219 127 15 of of ADP cana-1219 127 16 graphs graph NOUN cana-1219 127 17 , , PUNCT cana-1219 127 18 ph ph PROPN cana-1219 127 19 . . PUNCT cana-1219 128 1 d d X cana-1219 128 2 thesis thesis NOUN cana-1219 128 3 , , PUNCT cana-1219 128 4 2020 2020 NUM cana-1219 129 1 [ [ X cana-1219 129 2 5 5 NUM cana-1219 129 3 ] ] PUNCT cana-1219 129 4 maria maria PROPN cana-1219 129 5 chudnovsky chudnovsky PROPN cana-1219 129 6 , , PUNCT cana-1219 129 7 neil neil PROPN cana-1219 129 8 robertson robertson PROPN cana-1219 129 9 , , PUNCT cana-1219 129 10 paul paul PROPN cana-1219 129 11 seymour seymour PROPN cana-1219 129 12 , , PUNCT cana-1219 129 13 robin robin PROPN cana-1219 129 14 thomas thomas PROPN cana-1219 129 15 , , PUNCT cana-1219 129 16 the the DET cana-1219 129 17 strong strong ADJ cana-1219 129 18 perfect perfect ADJ cana-1219 129 19 graph graph NOUN cana-1219 129 20 theorem theorem NOUN cana-1219 129 21 , , PUNCT cana-1219 129 22 ann ann PROPN cana-1219 129 23 . . PROPN cana-1219 129 24 of of ADP cana-1219 129 25 math math NOUN cana-1219 129 26 , , PUNCT cana-1219 129 27 2006 2006 NUM cana-1219 129 28 . . PUNCT cana-1219 130 1 [ [ X cana-1219 130 2 6 6 NUM cana-1219 130 3 ] ] PUNCT cana-1219 130 4 t. t. PROPN cana-1219 130 5 hamada hamada PROPN cana-1219 130 6 and and CCONJ cana-1219 130 7 i. i. PROPN cana-1219 130 8 yoshimura yoshimura PROPN cana-1219 130 9 , , PUNCT cana-1219 130 10 traversability traversability NOUN cana-1219 130 11 and and CCONJ cana-1219 130 12 connectivity connectivity NOUN cana-1219 130 13 of of ADP cana-1219 130 14 the the DET cana-1219 130 15 middle middle ADJ cana-1219 130 16 graphs graph NOUN cana-1219 130 17 of of ADP cana-1219 130 18 a a DET cana-1219 130 19 graph graph NOUN cana-1219 130 20 , , PUNCT cana-1219 130 21 discrete discrete ADJ cana-1219 130 22 mathematics mathematic NOUN cana-1219 130 23 , , PUNCT cana-1219 130 24 14 14 NUM cana-1219 130 25 , , PUNCT cana-1219 130 26 1976 1976 NUM cana-1219 130 27 . . PUNCT cana-1219 131 1 [ [ X cana-1219 131 2 7 7 NUM cana-1219 131 3 ] ] X cana-1219 131 4 vernold vernold ADJ cana-1219 131 5 vivin vivin NOUN cana-1219 131 6 . . PUNCT cana-1219 132 1 j j PROPN cana-1219 132 2 , , PUNCT cana-1219 132 3 kowsalya kowsalya PROPN cana-1219 132 4 . . PUNCT cana-1219 133 1 v v NOUN cana-1219 133 2 , , PUNCT cana-1219 133 3 and and CCONJ cana-1219 133 4 vimal vimal PROPN cana-1219 133 5 kumar kumar PROPN cana-1219 133 6 . . PROPN cana-1219 134 1 s s PART cana-1219 134 2 , , PUNCT cana-1219 134 3 on on ADP cana-1219 134 4 star star ADJ cana-1219 134 5 chromatic chromatic ADJ cana-1219 134 6 number number NOUN cana-1219 134 7 of of ADP cana-1219 134 8 prism prism NOUN cana-1219 134 9 graph graph NOUN cana-1219 134 10 families family NOUN cana-1219 134 11 , , PUNCT cana-1219 134 12 twms twms PROPN cana-1219 134 13 journal journal PROPN cana-1219 134 14 of of ADP cana-1219 134 15 applied apply VERB cana-1219 134 16 and and CCONJ cana-1219 134 17 engineering engineering NOUN cana-1219 134 18 mathematics mathematic NOUN cana-1219 134 19 , , PUNCT cana-1219 134 20 vol.9 vol.9 PROPN cana-1219 134 21 , , PUNCT cana-1219 134 22 no.3 no.3 NOUN cana-1219 134 23 687692 687692 NUM cana-1219 134 24 , , PUNCT cana-1219 134 25 2019 2019 NUM cana-1219 134 26 . . PUNCT cana-1219 135 1 https://shodhganga.inflibnet.ac.in/jspui/handle/10603/466685 https://shodhganga.inflibnet.ac.in/jspui/handle/10603/466685 VERB