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PUNCT iajs-2436 39 1 an an DET iajs-2436 39 2 edge edge NOUN iajs-2436 39 3 whose whose DET iajs-2436 39 4 end end NOUN iajs-2436 39 5 - - PUNCT iajs-2436 39 6 vertices vertex NOUN iajs-2436 39 7 are be AUX iajs-2436 39 8 the the DET iajs-2436 39 9 same same ADJ iajs-2436 39 10 is be AUX iajs-2436 39 11 called call VERB iajs-2436 39 12 loop loop NOUN iajs-2436 39 13 . . PUNCT iajs-2436 40 1 definition definition NOUN iajs-2436 40 2 5 5 NUM iajs-2436 40 3 : : PUNCT iajs-2436 40 4 [ [ X iajs-2436 40 5 8 8 NUM iajs-2436 40 6 ] ] PUNCT iajs-2436 40 7 . . PUNCT iajs-2436 41 1 if if SCONJ iajs-2436 41 2 there there PRON iajs-2436 41 3 are be VERB iajs-2436 41 4 more more ADJ iajs-2436 41 5 than than ADP iajs-2436 41 6 one one NUM iajs-2436 41 7 edges edge NOUN iajs-2436 41 8 associated associate VERB iajs-2436 41 9 with with ADP iajs-2436 41 10 a a DET iajs-2436 41 11 given give VERB iajs-2436 41 12 pair pair NOUN iajs-2436 41 13 of of ADP iajs-2436 41 14 vertices vertex NOUN iajs-2436 41 15 , , PUNCT iajs-2436 41 16 then then ADV iajs-2436 41 17 these these DET iajs-2436 41 18 edges edge NOUN iajs-2436 41 19 are be AUX iajs-2436 41 20 called call VERB iajs-2436 41 21 multiple multiple ADJ iajs-2436 41 22 edges edge NOUN iajs-2436 41 23 or or CCONJ iajs-2436 41 24 parallel parallel ADJ iajs-2436 41 25 edges edge NOUN iajs-2436 41 26 . . PUNCT iajs-2436 42 1 definition definition NOUN iajs-2436 42 2 6 6 NUM iajs-2436 42 3 : : PUNCT iajs-2436 42 4 [ [ X iajs-2436 42 5 8 8 NUM iajs-2436 42 6 ] ] PUNCT iajs-2436 42 7 . . PUNCT iajs-2436 43 1 a a DET iajs-2436 43 2 simple simple ADJ iajs-2436 43 3 graph graph NOUN iajs-2436 43 4 that that PRON iajs-2436 43 5 has have VERB iajs-2436 43 6 no no DET iajs-2436 43 7 self self NOUN iajs-2436 43 8 - - PUNCT iajs-2436 43 9 loops loop NOUN iajs-2436 43 10 or or CCONJ iajs-2436 43 11 multiple multiple ADJ iajs-2436 43 12 edges edge NOUN iajs-2436 43 13 . . PUNCT iajs-2436 44 1 definition definition NOUN iajs-2436 44 2 7 7 NUM iajs-2436 44 3 : : PUNCT iajs-2436 45 1 [ [ X iajs-2436 45 2 8 8 NUM iajs-2436 45 3 ] ] PUNCT iajs-2436 45 4 . . PUNCT iajs-2436 46 1 a a DET iajs-2436 46 2 graph graph NOUN iajs-2436 46 3 h h NOUN iajs-2436 46 4 is be AUX iajs-2436 46 5 said say VERB iajs-2436 46 6 to to PART iajs-2436 46 7 be be AUX iajs-2436 46 8 a a DET iajs-2436 46 9 subgraph subgraph NOUN iajs-2436 46 10 of of ADP iajs-2436 46 11 a a DET iajs-2436 46 12 graph graph NOUN iajs-2436 46 13 g g NOUN iajs-2436 46 14 if if SCONJ iajs-2436 46 15 all all DET iajs-2436 46 16 the the DET iajs-2436 46 17 edges edge NOUN iajs-2436 46 18 and and CCONJ iajs-2436 46 19 all all DET iajs-2436 46 20 the the DET iajs-2436 46 21 vertices vertex NOUN iajs-2436 46 22 of of ADP iajs-2436 46 23 h h NOUN iajs-2436 46 24 are be AUX iajs-2436 46 25 in in ADP iajs-2436 46 26 g g PROPN iajs-2436 46 27 , , PUNCT iajs-2436 46 28 and and CCONJ iajs-2436 46 29 each each DET iajs-2436 46 30 edge edge NOUN iajs-2436 46 31 of of ADP iajs-2436 46 32 h h NOUN iajs-2436 46 33 has have VERB iajs-2436 46 34 the the DET iajs-2436 46 35 same same ADJ iajs-2436 46 36 end end NOUN iajs-2436 46 37 vertices vertex NOUN iajs-2436 46 38 in in ADP iajs-2436 46 39 h h NOUN iajs-2436 46 40 as as ADP iajs-2436 46 41 in in ADP iajs-2436 46 42 g g PROPN iajs-2436 46 43 . . PUNCT iajs-2436 47 1 definition definition NOUN iajs-2436 47 2 8 8 NUM iajs-2436 47 3 : : PUNCT iajs-2436 48 1 [ [ X iajs-2436 48 2 9 9 NUM iajs-2436 48 3 ] ] PUNCT iajs-2436 48 4 . . PUNCT iajs-2436 49 1 a a DET iajs-2436 49 2 path path NOUN iajs-2436 49 3 is be AUX iajs-2436 49 4 a a DET iajs-2436 49 5 graph graph NOUN iajs-2436 49 6 g g NOUN iajs-2436 49 7 that that PRON iajs-2436 49 8 contains contain VERB iajs-2436 49 9 a a DET iajs-2436 49 10 list𝑣 list𝑣 NOUN iajs-2436 49 11 , , PUNCT iajs-2436 49 12 𝑣 𝑣 X iajs-2436 49 13 , , PUNCT iajs-2436 49 14 … … PUNCT iajs-2436 49 15 , , PUNCT iajs-2436 49 16 𝑣 𝑣 PRON iajs-2436 49 17 of of ADP iajs-2436 49 18 vertices vertex NOUN iajs-2436 49 19 of of ADP iajs-2436 49 20 g g PROPN iajs-2436 49 21 s.t s.t PROPN iajs-2436 49 22 . . PROPN iajs-2436 50 1 for for ADP iajs-2436 50 2 1 1 NUM iajs-2436 50 3 𝑖 𝑖 SYM iajs-2436 50 4 𝑝 𝑝 PROPN iajs-2436 50 5 1 1 NUM iajs-2436 50 6 , , PUNCT iajs-2436 50 7 there there PRON iajs-2436 50 8 is be VERB iajs-2436 50 9 an an DET iajs-2436 50 10 edge edge NOUN iajs-2436 50 11 𝑣 𝑣 ADP iajs-2436 50 12 , , PUNCT iajs-2436 50 13 𝑣 𝑣 X iajs-2436 50 14 in in ADP iajs-2436 50 15 g g PROPN iajs-2436 50 16 and and CCONJ iajs-2436 50 17 these these PRON iajs-2436 50 18 are be AUX iajs-2436 50 19 the the DET iajs-2436 50 20 only only ADJ iajs-2436 50 21 edges edge NOUN iajs-2436 50 22 in in ADP iajs-2436 50 23 g g PROPN iajs-2436 50 24 . . PUNCT iajs-2436 51 1 definition definition NOUN iajs-2436 51 2 9 9 NUM iajs-2436 51 3 : : PUNCT iajs-2436 52 1 [ [ X iajs-2436 52 2 9 9 NUM iajs-2436 52 3 ] ] PUNCT iajs-2436 52 4 . . PUNCT iajs-2436 53 1 let let VERB iajs-2436 53 2 𝑣 𝑣 PRON iajs-2436 53 3 and and CCONJ iajs-2436 53 4 𝑣 𝑣 AUX iajs-2436 53 5 be be AUX iajs-2436 53 6 two two NUM iajs-2436 53 7 vertices vertex NOUN iajs-2436 53 8 , , PUNCT iajs-2436 53 9 d d X iajs-2436 53 10 ( ( PUNCT iajs-2436 53 11 𝑣 𝑣 NOUN iajs-2436 53 12 , , PUNCT iajs-2436 53 13 𝑣 𝑣 X iajs-2436 53 14 ) ) PUNCT iajs-2436 53 15 is be AUX iajs-2436 53 16 called call VERB iajs-2436 53 17 a a DET iajs-2436 53 18 distance distance NOUN iajs-2436 53 19 from from ADP iajs-2436 53 20 𝑣 𝑣 PRON iajs-2436 53 21 to to ADP iajs-2436 53 22 𝑣 𝑣 PRON iajs-2436 53 23 if if SCONJ iajs-2436 53 24 it it PRON iajs-2436 53 25 is be AUX iajs-2436 53 26 the the DET iajs-2436 53 27 shortest short ADJ iajs-2436 53 28 path path NOUN iajs-2436 53 29 from from ADP iajs-2436 53 30 𝑣 𝑣 PRON iajs-2436 53 31 to to ADP iajs-2436 53 32 𝑣 𝑣 PROPN iajs-2436 53 33 . . PUNCT iajs-2436 54 1 definition definition NOUN iajs-2436 54 2 10 10 NUM iajs-2436 54 3 : : PUNCT iajs-2436 55 1 [ [ X iajs-2436 55 2 9 9 NUM iajs-2436 55 3 ] ] PUNCT iajs-2436 55 4 . . PUNCT iajs-2436 56 1 a a DET iajs-2436 56 2 close close ADJ iajs-2436 56 3 path path NOUN iajs-2436 56 4 is be AUX iajs-2436 56 5 called call VERB iajs-2436 56 6 cylce cylce PROPN iajs-2436 56 7 , , PUNCT iajs-2436 56 8 the the DET iajs-2436 56 9 degree degree NOUN iajs-2436 56 10 of of ADP iajs-2436 56 11 each each DET iajs-2436 56 12 vertex vertex NOUN iajs-2436 56 13 of of ADP iajs-2436 56 14 a a DET iajs-2436 56 15 cycle cycle NOUN iajs-2436 56 16 graph graph NOUN iajs-2436 56 17 is be AUX iajs-2436 56 18 two two NUM iajs-2436 56 19 , , PUNCT iajs-2436 56 20 a a DET iajs-2436 56 21 cycle cycle NOUN iajs-2436 56 22 with with ADP iajs-2436 56 23 n n ADP iajs-2436 56 24 vertices vertex NOUN iajs-2436 56 25 denoted denote VERB iajs-2436 56 26 by by ADP iajs-2436 56 27 𝐶 𝐶 PROPN iajs-2436 56 28 . . PUNCT iajs-2436 57 1 definition definition NOUN iajs-2436 57 2 11 11 NUM iajs-2436 57 3 : : PUNCT iajs-2436 58 1 [ [ X iajs-2436 58 2 10 10 NUM iajs-2436 58 3 ] ] PUNCT iajs-2436 58 4 . . PUNCT iajs-2436 59 1 let let VERB iajs-2436 59 2 g g NOUN iajs-2436 59 3 ( ( PUNCT iajs-2436 59 4 v v NOUN iajs-2436 59 5 , , PUNCT iajs-2436 59 6 e e NOUN iajs-2436 59 7 ) ) PUNCT iajs-2436 59 8 be be AUX iajs-2436 59 9 a a DET iajs-2436 59 10 graph graph NOUN iajs-2436 59 11 and and CCONJ iajs-2436 59 12 c c NOUN iajs-2436 59 13 ⊂ ⊂ PROPN iajs-2436 59 14 g g PROPN iajs-2436 59 15 , , PUNCT iajs-2436 59 16 is be AUX iajs-2436 59 17 called call VERB iajs-2436 59 18 clique clique NOUN iajs-2436 59 19 if if SCONJ iajs-2436 59 20 the the DET iajs-2436 59 21 induced induced ADJ iajs-2436 59 22 sub sub NOUN iajs-2436 59 23 graph graph NOUN iajs-2436 59 24 of of ADP iajs-2436 59 25 g g NOUN iajs-2436 59 26 induced induce VERB iajs-2436 59 27 by by ADP iajs-2436 59 28 c c PROPN iajs-2436 59 29 is be AUX iajs-2436 59 30 a a DET iajs-2436 59 31 complete complete ADJ iajs-2436 59 32 graph graph NOUN iajs-2436 59 33 . . PUNCT iajs-2436 60 1 the the DET iajs-2436 60 2 clique clique NOUN iajs-2436 60 3 is be AUX iajs-2436 60 4 called call VERB iajs-2436 60 5 maximal maximal ADJ iajs-2436 60 6 if if SCONJ iajs-2436 60 7 there there PRON iajs-2436 60 8 is be VERB iajs-2436 60 9 no no DET iajs-2436 60 10 clique clique NOUN iajs-2436 60 11 with with ADP iajs-2436 60 12 more more ADJ iajs-2436 60 13 vertices vertex NOUN iajs-2436 60 14 . . PUNCT iajs-2436 61 1 definition definition NOUN iajs-2436 61 2 12 12 NUM iajs-2436 61 3 : : PUNCT iajs-2436 62 1 [ [ X iajs-2436 62 2 11 11 NUM iajs-2436 62 3 ] ] PUNCT iajs-2436 62 4 . . PUNCT iajs-2436 63 1 a a DET iajs-2436 63 2 h h NOUN iajs-2436 63 3 - - PUNCT iajs-2436 63 4 coloring coloring NOUN iajs-2436 63 5 of of ADP iajs-2436 63 6 the the DET iajs-2436 63 7 vertex vertex NOUN iajs-2436 63 8 set set NOUN iajs-2436 63 9 of of ADP iajs-2436 63 10 a a DET iajs-2436 63 11 graph graph NOUN iajs-2436 63 12 g g NOUN iajs-2436 63 13 is be AUX iajs-2436 63 14 a a DET iajs-2436 63 15 function function NOUN iajs-2436 63 16 𝛾 𝛾 ADP iajs-2436 63 17 : : PUNCT iajs-2436 63 18 v(g v(g ADJ iajs-2436 63 19 ) ) PUNCT iajs-2436 63 20 → → SYM iajs-2436 63 21 1,2 1,2 NUM iajs-2436 63 22 , , PUNCT iajs-2436 63 23 … … PUNCT iajs-2436 63 24 , , PUNCT iajs-2436 63 25 ℎ ℎ PROPN iajs-2436 63 26 such such ADJ iajs-2436 63 27 that that SCONJ iajs-2436 63 28 𝛾(𝑣 𝛾(𝑣 PROPN iajs-2436 63 29 ) ) PUNCT iajs-2436 63 30 𝛾(𝑣 𝛾(𝑣 PROPN iajs-2436 63 31 ) ) PUNCT iajs-2436 63 32 whenever whenever SCONJ iajs-2436 63 33 𝑣 𝑣 PRON iajs-2436 63 34 is be AUX iajs-2436 63 35 adjacent adjacent ADJ iajs-2436 63 36 to to ADP iajs-2436 63 37 𝑣 𝑣 ADP iajs-2436 63 38 , , PUNCT iajs-2436 63 39 if if SCONJ iajs-2436 63 40 a a DET iajs-2436 63 41 hcoloring hcoloring NOUN iajs-2436 63 42 of of ADP iajs-2436 63 43 g g PROPN iajs-2436 63 44 exists exist VERB iajs-2436 63 45 , , PUNCT iajs-2436 63 46 then then ADV iajs-2436 63 47 g g PROPN iajs-2436 63 48 is be AUX iajs-2436 63 49 called call VERB iajs-2436 63 50 hcolorable hcolorable ADJ iajs-2436 63 51 . . PUNCT iajs-2436 64 1 definition definition NOUN iajs-2436 64 2 13 13 NUM iajs-2436 64 3 : : PUNCT iajs-2436 65 1 [ [ X iajs-2436 65 2 11 11 NUM iajs-2436 65 3 ] ] PUNCT iajs-2436 65 4 . . PUNCT iajs-2436 66 1 the the DET iajs-2436 66 2 chromatic chromatic ADJ iajs-2436 66 3 number number NOUN iajs-2436 66 4 of of ADP iajs-2436 66 5 g g PROPN iajs-2436 66 6 is be AUX iajs-2436 66 7 defined define VERB iajs-2436 66 8 as as ADP iajs-2436 66 9 𝒳 𝒳 PROPN iajs-2436 66 10 ( ( PUNCT iajs-2436 66 11 𝐺 𝐺 NOUN iajs-2436 66 12 ) ) PUNCT iajs-2436 66 13 = = SYM iajs-2436 66 14 min min PROPN iajs-2436 66 15 { { PUNCT iajs-2436 66 16 h h NOUN iajs-2436 66 17 : : PUNCT iajs-2436 66 18 g g PROPN iajs-2436 66 19 is be AUX iajs-2436 66 20 hcolorable hcolorable ADJ iajs-2436 66 21 } } PUNCT iajs-2436 67 1 where where SCONJ iajs-2436 67 2 𝒳 𝒳 PROPN iajs-2436 67 3 ( ( PUNCT iajs-2436 67 4 𝐺 𝐺 NOUN iajs-2436 67 5 ) ) PUNCT iajs-2436 67 6 = = SYM iajs-2436 67 7 h h NOUN iajs-2436 67 8 , , PUNCT iajs-2436 67 9 g g PROPN iajs-2436 67 10 is be AUX iajs-2436 67 11 called call VERB iajs-2436 67 12 hchromatic hchromatic ADJ iajs-2436 67 13 . . PUNCT iajs-2436 68 1 theorem theorem VERB iajs-2436 68 2 14 14 NUM iajs-2436 69 1 [ [ SYM iajs-2436 69 2 10 10 NUM iajs-2436 69 3 ] ] PUNCT iajs-2436 69 4 . . PUNCT iajs-2436 70 1 for for ADP iajs-2436 70 2 circular circular ADJ iajs-2436 70 3 graph graph NOUN iajs-2436 70 4 𝐶 𝐶 PROPN iajs-2436 70 5 one one NOUN iajs-2436 70 6 has have VERB iajs-2436 70 7 𝒳 𝒳 PROPN iajs-2436 70 8 ( ( PUNCT iajs-2436 70 9 𝐶 𝐶 PROPN iajs-2436 70 10 ) ) PUNCT iajs-2436 70 11 = = SYM iajs-2436 70 12 2 2 NUM iajs-2436 70 13 𝑤ℎ𝑒𝑛 𝑤ℎ𝑒𝑛 VERB iajs-2436 70 14 𝑛 𝑛 DET iajs-2436 70 15 𝑖𝑠 𝑖𝑠 NOUN iajs-2436 71 1 𝑒𝑣𝑒𝑛 𝑒𝑣𝑒𝑛 ADJ iajs-2436 71 2 3 3 NUM iajs-2436 71 3 𝑤ℎ𝑒𝑛 𝑤ℎ𝑒𝑛 VERB iajs-2436 71 4 𝑛 𝑛 DET iajs-2436 71 5 𝑖𝑠 𝑖𝑠 NOUN iajs-2436 71 6 𝑜𝑑𝑑 𝑜𝑑𝑑 NOUN iajs-2436 71 7 in in ADP iajs-2436 71 8 other other ADJ iajs-2436 71 9 words word NOUN iajs-2436 71 10 𝒳 𝒳 PROPN iajs-2436 71 11 ( ( PUNCT iajs-2436 71 12 𝐶 𝐶 PROPN iajs-2436 71 13 ) ) PUNCT iajs-2436 71 14 = = SYM iajs-2436 71 15 3 3 NUM iajs-2436 71 16 , , PUNCT iajs-2436 71 17 𝒳 𝒳 PROPN iajs-2436 71 18 ( ( PUNCT iajs-2436 71 19 𝐶 𝐶 PROPN iajs-2436 71 20 ) ) PUNCT iajs-2436 71 21 = = SYM iajs-2436 71 22 2 2 NUM iajs-2436 71 23 for for ADP iajs-2436 71 24 i i PRON iajs-2436 71 25 ∈ ∈ PROPN iajs-2436 71 26 { { PUNCT iajs-2436 71 27 1,2 1,2 NUM iajs-2436 71 28 , , PUNCT iajs-2436 71 29 3 3 NUM iajs-2436 71 30 , , PUNCT iajs-2436 71 31 … … PUNCT iajs-2436 71 32 } } PUNCT iajs-2436 71 33 . . 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PUNCT iajs-2436 74 1 let let VERB iajs-2436 74 2 r r PRON iajs-2436 74 3 be be AUX iajs-2436 74 4 a a DET iajs-2436 74 5 ring ring NOUN iajs-2436 74 6 , , PUNCT iajs-2436 74 7 b(r b(r PROPN iajs-2436 74 8 ) ) PUNCT iajs-2436 74 9 = = PRON iajs-2436 74 10 { { PUNCT iajs-2436 74 11 ( ( PUNCT iajs-2436 74 12 a a DET iajs-2436 74 13 , , PUNCT iajs-2436 74 14 b b NOUN iajs-2436 74 15 ) ) PUNCT iajs-2436 74 16 : : PUNCT iajs-2436 74 17 arb arb PROPN iajs-2436 74 18 = = SYM iajs-2436 74 19 0 0 NUM iajs-2436 74 20 or or CCONJ iajs-2436 74 21 bra bra NOUN iajs-2436 74 22 = = SYM iajs-2436 74 23 0 0 NUM iajs-2436 74 24 , , PUNCT iajs-2436 74 25 a a PRON iajs-2436 74 26 , , PUNCT iajs-2436 74 27 b b NOUN iajs-2436 74 28 ∈r ∈r NOUN iajs-2436 74 29 , , PUNCT iajs-2436 74 30 a a DET iajs-2436 74 31 b b NOUN iajs-2436 74 32 , , PUNCT iajs-2436 74 33 a a DET iajs-2436 74 34 0 0 NUM iajs-2436 74 35 , , PUNCT iajs-2436 74 36 b b NOUN iajs-2436 74 37 0 0 NUM iajs-2436 74 38 } } PUNCT iajs-2436 74 39 . . PUNCT iajs-2436 75 1 then then ADV iajs-2436 75 2 χ χ DET iajs-2436 75 3 pg(r)= pg(r)= NOUN iajs-2436 75 4 χ χ X iajs-2436 75 5 g(b(r g(b(r PROPN iajs-2436 75 6 ) ) PUNCT iajs-2436 75 7 ) ) PUNCT iajs-2436 76 1 +1 +1 PROPN iajs-2436 76 2 , , PUNCT iajs-2436 76 3 when when SCONJ iajs-2436 76 4 g(b(r g(b(r PROPN iajs-2436 76 5 ) ) PUNCT iajs-2436 76 6 ) ) PUNCT iajs-2436 76 7 is be AUX iajs-2436 76 8 the the DET iajs-2436 76 9 induced induced ADJ iajs-2436 76 10 sub sub NOUN iajs-2436 76 11 graph graph NOUN iajs-2436 76 12 of of ADP iajs-2436 76 13 pg(r pg(r NOUN iajs-2436 76 14 ) ) PUNCT iajs-2436 76 15 whose whose DET iajs-2436 76 16 edges edge NOUN iajs-2436 76 17 are be AUX iajs-2436 76 18 elements element NOUN iajs-2436 76 19 of of ADP iajs-2436 76 20 b(r b(r NOUN iajs-2436 76 21 ) ) PUNCT iajs-2436 76 22 . . PUNCT iajs-2436 77 1 3 3 X iajs-2436 77 2 . . 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PUNCT iajs-2436 78 1 a a DET iajs-2436 78 2 graph graph NOUN iajs-2436 78 3 g g PROPN iajs-2436 78 4 ( ( PUNCT iajs-2436 78 5 v v NOUN iajs-2436 78 6 , , PUNCT iajs-2436 78 7 e e NOUN iajs-2436 78 8 ) ) PUNCT iajs-2436 78 9 is be AUX iajs-2436 78 10 said say VERB iajs-2436 78 11 to to PART iajs-2436 78 12 be be AUX iajs-2436 78 13 ( ( PUNCT iajs-2436 78 14 pseudo pseudo NOUN iajs-2436 78 15 - - PROPN iajs-2436 78 16 von von PROPN iajs-2436 78 17 neumann neumann PROPN iajs-2436 78 18 regular regular ADJ iajs-2436 78 19 graph graph NOUN iajs-2436 78 20 ) ) PUNCT iajs-2436 78 21 of of ADP iajs-2436 78 22 r r NOUN iajs-2436 78 23 if if SCONJ iajs-2436 78 24 v(g v(g ADJ iajs-2436 78 25 ) ) PUNCT iajs-2436 78 26 = = VERB iajs-2436 78 27 r r NOUN iajs-2436 78 28 and and CCONJ iajs-2436 78 29 e e NOUN iajs-2436 78 30 ( ( PUNCT iajs-2436 78 31 g g NOUN iajs-2436 78 32 ) ) PUNCT iajs-2436 78 33 = = NOUN iajs-2436 78 34 { { PUNCT iajs-2436 78 35 𝑎𝑏/ 𝑎𝑏/ ADV iajs-2436 78 36 𝑎 𝑎 ADP iajs-2436 78 37 𝑎 𝑎 NOUN iajs-2436 78 38 𝑏 𝑏 NOUN iajs-2436 78 39 or or CCONJ iajs-2436 78 40 𝑏 𝑏 NOUN iajs-2436 78 41 𝑏 𝑏 PROPN iajs-2436 78 42 𝑎 𝑎 NOUN iajs-2436 78 43 and and CCONJ iajs-2436 78 44 𝑎 𝑎 NOUN iajs-2436 78 45 𝑏 𝑏 NOUN iajs-2436 78 46 } } PUNCT iajs-2436 78 47 denoted denote VERB iajs-2436 78 48 by by ADP iajs-2436 78 49 p p NOUN iajs-2436 78 50 - - PUNCT iajs-2436 78 51 vg(r vg(r NOUN iajs-2436 78 52 ) ) PUNCT iajs-2436 78 53 , , PUNCT iajs-2436 78 54 shortly shortly ADV iajs-2436 78 55 p p PROPN iajs-2436 78 56 - - PUNCT iajs-2436 78 57 von von PROPN iajs-2436 78 58 neumann neumann PROPN iajs-2436 78 59 regular regular ADJ iajs-2436 78 60 graph graph NOUN iajs-2436 78 61 . . PUNCT iajs-2436 78 62     SPACE iajs-2436 79 1 151 151 NUM iajs-2436 79 2     SPACE iajs-2436 79 3 ibn ibn PROPN iajs-2436 79 4 al al PROPN iajs-2436 79 5 - - PUNCT iajs-2436 79 6 haitham haitham PROPN iajs-2436 79 7 jour jour X iajs-2436 79 8 . . PROPN iajs-2436 79 9 for for ADP iajs-2436 79 10 pure pure ADJ iajs-2436 79 11 & & CCONJ iajs-2436 79 12 appl appl PROPN iajs-2436 79 13 . . PUNCT iajs-2436 80 1 sci sci PROPN iajs-2436 80 2 . . PROPN iajs-2436 81 1 33 33 NUM iajs-2436 81 2 ( ( PUNCT iajs-2436 81 3 2 2 NUM iajs-2436 81 4 ) ) PUNCT iajs-2436 81 5 2020 2020 NUM iajs-2436 81 6 example example NOUN iajs-2436 81 7 3.2 3.2 NUM iajs-2436 81 8 𝑍 𝑍 NOUN iajs-2436 81 9 = = NOUN iajs-2436 81 10 { { PUNCT iajs-2436 81 11 0,1 0,1 NUM iajs-2436 81 12 } } PUNCT iajs-2436 81 13 figure figure NOUN iajs-2436 81 14 1 1 NUM iajs-2436 81 15 : : PUNCT iajs-2436 81 16 p p NUM iajs-2436 81 17 - - PUNCT iajs-2436 81 18 vg(𝑍 vg(𝑍 NOUN iajs-2436 81 19 ) ) PUNCT iajs-2436 81 20 𝑍 𝑍 PROPN iajs-2436 81 21 = = PRON iajs-2436 81 22 { { PUNCT iajs-2436 81 23 0,1,2 0,1,2 NOUN iajs-2436 81 24 } } PUNCT iajs-2436 81 25 figure figure NOUN iajs-2436 81 26 2 2 NUM iajs-2436 81 27 :p :p NOUN iajs-2436 81 28 - - PUNCT iajs-2436 81 29 vg(𝑍 vg(𝑍 ADJ iajs-2436 81 30 ) ) PUNCT iajs-2436 81 31 𝑍 𝑍 PROPN iajs-2436 81 32 = = PRON iajs-2436 81 33 { { PUNCT iajs-2436 81 34 0,1,2,3 0,1,2,3 NOUN iajs-2436 81 35 } } PUNCT iajs-2436 81 36 figure figure NOUN iajs-2436 81 37 3 3 NUM iajs-2436 81 38 :p :p NOUN iajs-2436 81 39 - - PUNCT iajs-2436 81 40 vg(𝑍 vg(𝑍 ADJ iajs-2436 81 41 ) ) PUNCT iajs-2436 81 42 𝑍 𝑍 PROPN iajs-2436 81 43 = = NOUN iajs-2436 81 44 { { PUNCT iajs-2436 81 45 0,1,2,3,4 0,1,2,3,4 ADJ iajs-2436 81 46 } } PUNCT iajs-2436 81 47 figure figure NOUN iajs-2436 81 48 4 4 NUM iajs-2436 81 49 :p :p ADJ iajs-2436 81 50 - - PUNCT iajs-2436 81 51 vg(𝑍 vg(𝑍 ADJ iajs-2436 81 52 ) ) PUNCT iajs-2436 81 53 note note VERB iajs-2436 81 54 3.3 3.3 NUM iajs-2436 81 55 c c NOUN iajs-2436 82 1 [ [ X iajs-2436 82 2 r r X iajs-2436 82 3 ] ] X iajs-2436 82 4 = = X iajs-2436 82 5 { { PUNCT iajs-2436 82 6 𝑎 𝑎 NOUN iajs-2436 82 7 , , PUNCT iajs-2436 82 8 𝑏 𝑏 PROPN iajs-2436 82 9 ∶ ∶ NOUN iajs-2436 82 10 𝑎 𝑎 PROPN iajs-2436 82 11 𝑏 𝑏 NOUN iajs-2436 82 12 , , PUNCT iajs-2436 82 13 𝑎 𝑎 PROPN iajs-2436 82 14 0 0 NUM iajs-2436 82 15 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-2436 82 16 𝑏 𝑏 PROPN iajs-2436 82 17 0 0 NUM iajs-2436 82 18 , , PUNCT iajs-2436 82 19 𝑎 𝑎 X iajs-2436 82 20 𝑎 𝑎 NOUN iajs-2436 82 21 𝑏 𝑏 NOUN iajs-2436 82 22 or or CCONJ iajs-2436 82 23 𝑏 𝑏 NOUN iajs-2436 82 24 𝑏 𝑏 PROPN iajs-2436 82 25 𝑎 𝑎 PROPN iajs-2436 82 26 } } PUNCT iajs-2436 82 27 ⊂ ⊂ PROPN iajs-2436 82 28 r r NOUN iajs-2436 82 29 r. r. PROPN iajs-2436 82 30 corollary corollary NOUN iajs-2436 82 31 3.4 3.4 NUM iajs-2436 82 32 ithe ithe ADJ iajs-2436 82 33 number number NOUN iajs-2436 82 34 of of ADP iajs-2436 82 35 element element NOUN iajs-2436 82 36 in in ADP iajs-2436 82 37 c c PROPN iajs-2436 83 1 [ [ X iajs-2436 83 2 r r X iajs-2436 83 3 ] ] X iajs-2436 83 4 is be AUX iajs-2436 83 5 less less ADJ iajs-2436 83 6 than than ADP iajs-2436 83 7 or or CCONJ iajs-2436 83 8 equal equal ADJ iajs-2436 83 9 to to ADP iajs-2436 83 10 number number NOUN iajs-2436 83 11 of of ADP iajs-2436 83 12 cycle cycle NOUN iajs-2436 83 13 𝐶 𝐶 PROPN iajs-2436 83 14 in in ADP iajs-2436 83 15 p p NOUN iajs-2436 83 16 - - PUNCT iajs-2436 83 17 vg(r vg(r NOUN iajs-2436 83 18 ) ) PUNCT iajs-2436 83 19 . . PUNCT iajs-2436 84 1 iiif iiif PROPN iajs-2436 84 2 c c PROPN iajs-2436 85 1 [ [ X iajs-2436 85 2 r r X iajs-2436 85 3 ] ] X iajs-2436 85 4 ∅ ∅ NOUN iajs-2436 85 5 , , PUNCT iajs-2436 85 6 then then ADV iajs-2436 85 7 the the DET iajs-2436 85 8 longest long ADJ iajs-2436 85 9 trail trail NOUN iajs-2436 85 10 has have AUX iajs-2436 85 11 length length NOUN iajs-2436 85 12 3 3 NUM iajs-2436 85 13 . . PUNCT iajs-2436 85 14 iiic iiic PROPN iajs-2436 86 1 [ [ X iajs-2436 86 2 r r X iajs-2436 86 3 ] ] X iajs-2436 86 4 r r NOUN iajs-2436 86 5 r. r. NOUN iajs-2436 86 6 proof proof NOUN iajs-2436 86 7 ilet ilet NOUN iajs-2436 86 8 ( ( PUNCT iajs-2436 86 9 a a PRON iajs-2436 86 10 , , PUNCT iajs-2436 86 11 b b NOUN iajs-2436 86 12 ) ) PUNCT iajs-2436 86 13 ∈ ∈ NOUN iajs-2436 86 14 c c NOUN iajs-2436 87 1 [ [ X iajs-2436 87 2 r r X iajs-2436 87 3 ] ] X iajs-2436 87 4 , , PUNCT iajs-2436 87 5 then then ADV iajs-2436 87 6 { { PUNCT iajs-2436 87 7 0 0 NUM iajs-2436 87 8 , , PUNCT iajs-2436 87 9 a a DET iajs-2436 87 10 , , PUNCT iajs-2436 87 11 b b NOUN iajs-2436 87 12 } } PUNCT iajs-2436 87 13 form form VERB iajs-2436 87 14 a a DET iajs-2436 87 15 cycle cycle NOUN iajs-2436 87 16 𝐶 𝐶 PROPN iajs-2436 87 17 in in ADP iajs-2436 87 18 p p NOUN iajs-2436 87 19 - - PUNCT iajs-2436 87 20 vg(r vg(r NOUN iajs-2436 87 21 ) ) PUNCT iajs-2436 87 22 , , PUNCT iajs-2436 87 23 if if SCONJ iajs-2436 87 24 ( ( PUNCT iajs-2436 87 25 a a DET iajs-2436 87 26 , , PUNCT iajs-2436 87 27 b b NOUN iajs-2436 87 28 ) ) PUNCT iajs-2436 87 29 , , PUNCT iajs-2436 87 30 ( ( PUNCT iajs-2436 87 31 s s X iajs-2436 87 32 , , PUNCT iajs-2436 87 33 t t PROPN iajs-2436 87 34 ) ) PUNCT iajs-2436 87 35 ∈ ∈ PROPN iajs-2436 87 36 c c NOUN iajs-2436 88 1 [ [ X iajs-2436 88 2 r r X iajs-2436 88 3 ] ] X iajs-2436 88 4 such such ADJ iajs-2436 88 5 that that SCONJ iajs-2436 88 6 ( ( PUNCT iajs-2436 88 7 a a DET iajs-2436 88 8 , , PUNCT iajs-2436 88 9 b b NOUN iajs-2436 88 10 ) ) PUNCT iajs-2436 88 11 ( ( PUNCT iajs-2436 88 12 s s PROPN iajs-2436 88 13 , , PUNCT iajs-2436 88 14 t t PROPN iajs-2436 88 15 ) ) PUNCT iajs-2436 88 16 then then ADV iajs-2436 88 17 a a DET iajs-2436 88 18 s s NOUN iajs-2436 88 19 or or CCONJ iajs-2436 88 20 b b NOUN iajs-2436 88 21 t t NOUN iajs-2436 88 22 and and CCONJ iajs-2436 88 23 so so ADV iajs-2436 88 24 the the DET iajs-2436 88 25 cycle cycle NOUN iajs-2436 88 26 𝐶 𝐶 PROPN iajs-2436 88 27 { { PUNCT iajs-2436 88 28 0 0 NUM iajs-2436 88 29 , , PUNCT iajs-2436 88 30 a a PRON iajs-2436 88 31 , , PUNCT iajs-2436 88 32 b b NOUN iajs-2436 88 33 } } PUNCT iajs-2436 88 34 and and CCONJ iajs-2436 88 35 { { PUNCT iajs-2436 88 36 0 0 NUM iajs-2436 88 37 , , PUNCT iajs-2436 88 38 s s X iajs-2436 88 39 , , PUNCT iajs-2436 88 40 t t NOUN iajs-2436 88 41 } } PUNCT iajs-2436 88 42 in in ADP iajs-2436 88 43 p p PROPN iajs-2436 88 44 - - PUNCT iajs-2436 88 45 vg(r vg(r VERB iajs-2436 88 46 ) ) PUNCT iajs-2436 88 47 are be AUX iajs-2436 88 48 distinct distinct ADJ iajs-2436 88 49 , , PUNCT iajs-2436 88 50 this this PRON iajs-2436 88 51 show show VERB iajs-2436 88 52 the the DET iajs-2436 88 53 number number NOUN iajs-2436 88 54 of of ADP iajs-2436 88 55 elements element NOUN iajs-2436 88 56 in in ADP iajs-2436 88 57 c c NOUN iajs-2436 88 58 [ [ X iajs-2436 88 59 r r X iajs-2436 88 60 ] ] X iajs-2436 88 61 is be AUX iajs-2436 88 62 less less ADJ iajs-2436 88 63 than than ADP iajs-2436 88 64 or or CCONJ iajs-2436 88 65 equal equal VERB iajs-2436 88 66 the the DET iajs-2436 88 67 number number NOUN iajs-2436 88 68 of of ADP iajs-2436 88 69 cycle cycle NOUN iajs-2436 88 70 𝐶 𝐶 PROPN iajs-2436 88 71 in in ADP iajs-2436 88 72 p p NOUN iajs-2436 88 73 - - PUNCT iajs-2436 88 74 vg(r vg(r NOUN iajs-2436 88 75 ) ) PUNCT iajs-2436 88 76 . . PUNCT iajs-2436 89 1 if if SCONJ iajs-2436 89 2 a a DET iajs-2436 89 3 = = NOUN iajs-2436 89 4 s s X iajs-2436 89 5 or or CCONJ iajs-2436 89 6 b b X iajs-2436 89 7 = = NOUN iajs-2436 89 8 t t X iajs-2436 89 9 then then ADV iajs-2436 89 10 the the DET iajs-2436 89 11 cycles cycle NOUN iajs-2436 89 12 { { PUNCT iajs-2436 89 13 0 0 NUM iajs-2436 89 14 , , PUNCT iajs-2436 89 15 a a DET iajs-2436 89 16 , , PUNCT iajs-2436 89 17 b b NOUN iajs-2436 89 18 } } PUNCT iajs-2436 89 19 , , PUNCT iajs-2436 89 20 { { PUNCT iajs-2436 89 21 0 0 NUM iajs-2436 89 22 , , PUNCT iajs-2436 89 23 s s X iajs-2436 89 24 , , PUNCT iajs-2436 89 25 t t PROPN iajs-2436 89 26 } } PUNCT iajs-2436 89 27 are be AUX iajs-2436 89 28 adjust adjust VERB iajs-2436 89 29 . . PUNCT iajs-2436 90 1 0 0 NUM iajs-2436 90 2 1 1 NUM iajs-2436 90 3 0 0 NUM iajs-2436 90 4 1 1 NUM iajs-2436 90 5 2 2 NUM iajs-2436 90 6 0 0 NUM iajs-2436 90 7 1 1 NUM iajs-2436 90 8 2 2 NUM iajs-2436 90 9 3 3 NUM iajs-2436 90 10 0 0 NUM iajs-2436 90 11 1 1 NUM iajs-2436 90 12 2 2 NUM iajs-2436 90 13 3 3 NUM iajs-2436 90 14 4 4 NUM iajs-2436 90 15     SPACE iajs-2436 90 16 152 152 NUM iajs-2436 90 17     SPACE iajs-2436 90 18 ibn ibn PROPN iajs-2436 90 19 al al PROPN iajs-2436 90 20 - - PUNCT iajs-2436 90 21 haitham haitham PROPN iajs-2436 90 22 jour jour X iajs-2436 90 23 . . PROPN iajs-2436 90 24 for for ADP iajs-2436 90 25 pure pure ADJ iajs-2436 90 26 & & CCONJ iajs-2436 90 27 appl appl PROPN iajs-2436 90 28 . . PUNCT iajs-2436 91 1 sci sci PROPN iajs-2436 91 2 . . PROPN iajs-2436 92 1 33 33 NUM iajs-2436 92 2 ( ( PUNCT iajs-2436 92 3 2 2 NUM iajs-2436 92 4 ) ) PUNCT iajs-2436 92 5 2020 2020 NUM iajs-2436 92 6 iilet iilet NOUN iajs-2436 92 7 ( ( PUNCT iajs-2436 92 8 a a PRON iajs-2436 92 9 , , PUNCT iajs-2436 92 10 b b NOUN iajs-2436 92 11 ) ) PUNCT iajs-2436 92 12 ∈ ∈ NOUN iajs-2436 92 13 c c NOUN iajs-2436 93 1 [ [ X iajs-2436 93 2 r],then r],then X iajs-2436 93 3 0a 0a PROPN iajs-2436 93 4 , , PUNCT iajs-2436 93 5 ab ab PROPN iajs-2436 93 6 , , PUNCT iajs-2436 93 7 0b 0b PROPN iajs-2436 93 8 is be AUX iajs-2436 93 9 cycle cycle NOUN iajs-2436 93 10 𝐶 𝐶 PROPN iajs-2436 93 11 , , PUNCT iajs-2436 93 12 this this DET iajs-2436 93 13 trail trail NOUN iajs-2436 93 14 is be AUX iajs-2436 93 15 of of ADP iajs-2436 93 16 length length NOUN iajs-2436 93 17 equal equal ADJ iajs-2436 93 18 to to ADP iajs-2436 93 19 3 3 NUM iajs-2436 93 20 hence hence ADV iajs-2436 93 21 , , PUNCT iajs-2436 93 22 the the DET iajs-2436 93 23 longest long ADJ iajs-2436 93 24 trail trail NOUN iajs-2436 93 25 has have VERB iajs-2436 93 26 length length NOUN iajs-2436 93 27 3 3 NUM iajs-2436 93 28 . . PUNCT iajs-2436 94 1 iiisince iiisince NOUN iajs-2436 94 2 ( ( PUNCT iajs-2436 94 3 0,a 0,a NOUN iajs-2436 94 4 ) ) PUNCT iajs-2436 94 5 ∈ ∈ PROPN iajs-2436 94 6 𝑅 𝑅 PROPN iajs-2436 94 7 𝑅 𝑅 PROPN iajs-2436 94 8 and and CCONJ iajs-2436 94 9 ( ( PUNCT iajs-2436 94 10 0,a 0,a PROPN iajs-2436 94 11 ) ) PUNCT iajs-2436 94 12 ∉ ∉ PROPN iajs-2436 94 13 c c PROPN iajs-2436 95 1 [ [ X iajs-2436 95 2 r r X iajs-2436 95 3 ] ] PUNCT iajs-2436 95 4 , , PUNCT iajs-2436 95 5 then then ADV iajs-2436 95 6 we we PRON iajs-2436 95 7 have have VERB iajs-2436 95 8 c c NOUN iajs-2436 96 1 [ [ X iajs-2436 96 2 r r X iajs-2436 96 3 ] ] X iajs-2436 96 4 ⊂ ⊂ PROPN iajs-2436 96 5 𝑅 𝑅 PROPN iajs-2436 96 6 𝑅 𝑅 PROPN iajs-2436 96 7 and and CCONJ iajs-2436 96 8 c c NOUN iajs-2436 97 1 [ [ X iajs-2436 97 2 r r X iajs-2436 97 3 ] ] X iajs-2436 97 4 𝑅 𝑅 PROPN iajs-2436 97 5 𝑅 𝑅 PROPN iajs-2436 97 6 . . PUNCT iajs-2436 98 1 by by ADP iajs-2436 98 2 the the DET iajs-2436 98 3 same same ADJ iajs-2436 98 4 way way NOUN iajs-2436 98 5 of of ADP iajs-2436 98 6 ( ( PUNCT iajs-2436 98 7 theorem theorem NOUN iajs-2436 98 8 2.15 2.15 NUM iajs-2436 98 9 ) ) PUNCT iajs-2436 98 10 we we PRON iajs-2436 98 11 can can AUX iajs-2436 98 12 prove prove VERB iajs-2436 98 13 below below ADP iajs-2436 98 14 theorem theorem PROPN iajs-2436 98 15 . . PUNCT iajs-2436 99 1 theorem theorem VERB iajs-2436 99 2 3.5 3.5 NUM iajs-2436 99 3 let let VERB iajs-2436 99 4 r r PRON iajs-2436 99 5 be be AUX iajs-2436 99 6 a a DET iajs-2436 99 7 ring ring NOUN iajs-2436 99 8 and and CCONJ iajs-2436 99 9 let let VERB iajs-2436 99 10 c c NOUN iajs-2436 99 11 [ [ X iajs-2436 99 12 r r X iajs-2436 99 13 ] ] X iajs-2436 99 14 = = X iajs-2436 99 15 { { PUNCT iajs-2436 99 16 𝑎 𝑎 NOUN iajs-2436 99 17 , , PUNCT iajs-2436 99 18 𝑏 𝑏 PROPN iajs-2436 99 19 ∶ ∶ NOUN iajs-2436 99 20 𝑎 𝑎 PROPN iajs-2436 99 21 𝑏 𝑏 NOUN iajs-2436 99 22 , , PUNCT iajs-2436 99 23 𝑎 𝑎 PROPN iajs-2436 99 24 0 0 NUM iajs-2436 99 25 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-2436 99 26 𝑏 𝑏 PROPN iajs-2436 99 27 0 0 NUM iajs-2436 99 28 , , PUNCT iajs-2436 99 29 𝑎 𝑎 X iajs-2436 99 30 𝑎 𝑎 NOUN iajs-2436 99 31 𝑏 𝑏 NOUN iajs-2436 99 32 or or CCONJ iajs-2436 99 33 𝑏 𝑏 NOUN iajs-2436 100 1 𝑏 𝑏 PROPN iajs-2436 100 2 𝑎 𝑎 PROPN iajs-2436 100 3 } } PUNCT iajs-2436 100 4 then then ADV iajs-2436 100 5 𝒳 𝒳 PROPN iajs-2436 100 6 ( ( PUNCT iajs-2436 100 7 p p NOUN iajs-2436 100 8 - - PUNCT iajs-2436 100 9 vg(r vg(r NOUN iajs-2436 100 10 ) ) PUNCT iajs-2436 100 11 ) ) PUNCT iajs-2436 100 12 = = SYM iajs-2436 100 13 𝒳 𝒳 PROPN iajs-2436 100 14 ( ( PUNCT iajs-2436 100 15 g(c g(c VERB iajs-2436 100 16 [ [ X iajs-2436 100 17 r r X iajs-2436 100 18 ] ] PUNCT iajs-2436 100 19 ) ) PUNCT iajs-2436 101 1 + + CCONJ iajs-2436 101 2 1 1 NUM iajs-2436 101 3 , , PUNCT iajs-2436 101 4 where where SCONJ iajs-2436 101 5 g(c g(c VERB iajs-2436 101 6 [ [ X iajs-2436 101 7 r r X iajs-2436 101 8 ] ] PUNCT iajs-2436 101 9 ) ) PUNCT iajs-2436 101 10 is be AUX iajs-2436 101 11 sub sub NOUN iajs-2436 101 12 graph graph NOUN iajs-2436 101 13 of of ADP iajs-2436 101 14 p p NOUN iajs-2436 101 15 - - PUNCT iajs-2436 101 16 vg(r vg(r NOUN iajs-2436 101 17 ) ) PUNCT iajs-2436 101 18 . . PUNCT iajs-2436 102 1 proof proof NOUN iajs-2436 102 2 let let VERB iajs-2436 102 3 r r NOUN iajs-2436 102 4 be be AUX iajs-2436 102 5 a a DET iajs-2436 102 6 ring ring NOUN iajs-2436 102 7 , , PUNCT iajs-2436 102 8 let let VERB iajs-2436 102 9 c c NOUN iajs-2436 102 10 [ [ X iajs-2436 102 11 r r X iajs-2436 102 12 ] ] X iajs-2436 102 13 = = X iajs-2436 102 14 { { PUNCT iajs-2436 102 15 𝑎 𝑎 NOUN iajs-2436 102 16 , , PUNCT iajs-2436 102 17 𝑏 𝑏 PROPN iajs-2436 102 18 ∶ ∶ NOUN iajs-2436 102 19 𝑎 𝑎 PROPN iajs-2436 102 20 𝑏 𝑏 NOUN iajs-2436 102 21 , , PUNCT iajs-2436 102 22 𝑎 𝑎 PROPN iajs-2436 102 23 0 0 NUM iajs-2436 102 24 𝑎𝑛𝑑 𝑎𝑛𝑑 ADJ iajs-2436 102 25 𝑏 𝑏 PROPN iajs-2436 102 26 0 0 NUM iajs-2436 102 27 , , PUNCT iajs-2436 102 28 𝑎 𝑎 X iajs-2436 102 29 𝑎 𝑎 NOUN iajs-2436 102 30 𝑏 𝑏 NOUN iajs-2436 102 31 or or CCONJ iajs-2436 102 32 𝑏 𝑏 NOUN iajs-2436 103 1 𝑏 𝑏 NOUN iajs-2436 103 2 𝑎 𝑎 PROPN iajs-2436 103 3 } } PUNCT iajs-2436 103 4 if if SCONJ iajs-2436 103 5 c c PROPN iajs-2436 104 1 [ [ X iajs-2436 104 2 r r X iajs-2436 104 3 ] ] X iajs-2436 104 4 = = SYM iajs-2436 104 5 ∅ ∅ NOUN iajs-2436 104 6 , , PUNCT iajs-2436 104 7 then then ADV iajs-2436 104 8 p p X iajs-2436 104 9 - - PUNCT iajs-2436 104 10 vg(r vg(r NOUN iajs-2436 104 11 ) ) PUNCT iajs-2436 104 12 graph graph NOUN iajs-2436 104 13 is be AUX iajs-2436 104 14 a a DET iajs-2436 104 15 star star NOUN iajs-2436 104 16 graph graph NOUN iajs-2436 104 17 and and CCONJ iajs-2436 104 18 𝒳 𝒳 PROPN iajs-2436 104 19 ( ( PUNCT iajs-2436 104 20 p p PROPN iajs-2436 104 21 - - PUNCT iajs-2436 104 22 vg(r vg(r NOUN iajs-2436 104 23 ) ) PUNCT iajs-2436 104 24 ) ) PUNCT iajs-2436 105 1 = = SYM iajs-2436 105 2 2 2 NUM iajs-2436 105 3 suppose suppose VERB iajs-2436 105 4 c c NOUN iajs-2436 105 5 [ [ X iajs-2436 105 6 r r X iajs-2436 105 7 ] ] X iajs-2436 105 8 ∅ ∅ NOUN iajs-2436 105 9 , , PUNCT iajs-2436 105 10 let let VERB iajs-2436 105 11 |𝐶 |𝐶 NOUN iajs-2436 105 12 𝑅 𝑅 PROPN iajs-2436 105 13 |= |= PUNCT iajs-2436 105 14 n n PRON iajs-2436 105 15 case case NOUN iajs-2436 105 16 i i PRON iajs-2436 105 17 : : PUNCT iajs-2436 105 18 let let VERB iajs-2436 105 19 g(c g(c VERB iajs-2436 105 20 [ [ X iajs-2436 105 21 r r NOUN iajs-2436 105 22 ] ] X iajs-2436 105 23 ) ) PUNCT iajs-2436 105 24 = = PUNCT iajs-2436 106 1 𝑃 𝑃 NOUN iajs-2436 106 2 then then ADV iajs-2436 106 3 𝒳 𝒳 PROPN iajs-2436 106 4 ( ( PUNCT iajs-2436 106 5 g(c g(c VERB iajs-2436 106 6 [ [ X iajs-2436 106 7 r r X iajs-2436 106 8 ] ] PUNCT iajs-2436 106 9 ) ) PUNCT iajs-2436 106 10 is be AUX iajs-2436 106 11 equal equal ADJ iajs-2436 106 12 to to ADP iajs-2436 106 13 2 2 NUM iajs-2436 106 14 , , PUNCT iajs-2436 106 15 since since SCONJ iajs-2436 106 16 0 0 NUM iajs-2436 106 17 is be AUX iajs-2436 106 18 adjacent adjacent ADJ iajs-2436 106 19 to to ADP iajs-2436 106 20 all all DET iajs-2436 106 21 vertices vertex NOUN iajs-2436 106 22 in in ADP iajs-2436 106 23 pvg(r pvg(r PROPN iajs-2436 106 24 ) ) PUNCT iajs-2436 106 25 , , PUNCT iajs-2436 106 26 so so ADV iajs-2436 106 27 we we PRON iajs-2436 106 28 have have VERB iajs-2436 106 29 a a DET iajs-2436 106 30 third third ADJ iajs-2436 106 31 color color NOUN iajs-2436 106 32 to to ADP iajs-2436 106 33 a a DET iajs-2436 106 34 vertex vertex NOUN iajs-2436 106 35 0 0 NUM iajs-2436 106 36 , , PUNCT iajs-2436 106 37 now now ADV iajs-2436 106 38 the the DET iajs-2436 106 39 vertices vertex NOUN iajs-2436 106 40 do do AUX iajs-2436 106 41 not not PART iajs-2436 106 42 belong belong VERB iajs-2436 106 43 to to ADP iajs-2436 106 44 𝑃 𝑃 PROPN iajs-2436 106 45 will will AUX iajs-2436 106 46 be be AUX iajs-2436 106 47 associated associate VERB iajs-2436 106 48 only only ADV iajs-2436 106 49 with with ADP iajs-2436 106 50 vertex vertex NOUN iajs-2436 106 51 0 0 NUM iajs-2436 106 52 , , PUNCT iajs-2436 106 53 this this DET iajs-2436 106 54 vertices vertex NOUN iajs-2436 106 55 can can AUX iajs-2436 106 56 colored color VERB iajs-2436 106 57 by by ADP iajs-2436 106 58 any any DET iajs-2436 106 59 color color NOUN iajs-2436 106 60 we we PRON iajs-2436 106 61 used use VERB iajs-2436 106 62 it it PRON iajs-2436 106 63 in in ADP iajs-2436 106 64 𝑃 𝑃 NOUN iajs-2436 106 65 . . PUNCT iajs-2436 107 1 hence hence ADV iajs-2436 107 2 𝒳 𝒳 PROPN iajs-2436 107 3 ( ( PUNCT iajs-2436 107 4 p p NOUN iajs-2436 107 5 - - PUNCT iajs-2436 107 6 vg(r vg(r NOUN iajs-2436 107 7 ) ) PUNCT iajs-2436 107 8 ) ) PUNCT iajs-2436 108 1 = = SYM iajs-2436 108 2 3 3 NUM iajs-2436 108 3 = = SYM iajs-2436 108 4 𝒳 𝒳 PROPN iajs-2436 108 5 ( ( PUNCT iajs-2436 108 6 g(c g(c VERB iajs-2436 108 7 [ [ X iajs-2436 108 8 r r X iajs-2436 108 9 ] ] PUNCT iajs-2436 108 10 ) ) PUNCT iajs-2436 108 11 +1 +1 PROPN iajs-2436 108 12 case case NOUN iajs-2436 108 13 ii ii NOUN iajs-2436 108 14 : : PUNCT iajs-2436 108 15 now now ADV iajs-2436 108 16 if if SCONJ iajs-2436 108 17 g(c g(c VERB iajs-2436 108 18 [ [ X iajs-2436 108 19 r r NOUN iajs-2436 108 20 ] ] X iajs-2436 108 21 ) ) PUNCT iajs-2436 108 22 = = SYM iajs-2436 108 23 𝐶 𝐶 PROPN iajs-2436 108 24 then then ADV iajs-2436 108 25 𝒳 𝒳 PROPN iajs-2436 108 26 ( ( PUNCT iajs-2436 108 27 g(c g(c VERB iajs-2436 108 28 [ [ X iajs-2436 108 29 r r X iajs-2436 108 30 ] ] X iajs-2436 108 31 ) ) PUNCT iajs-2436 108 32 ) ) PUNCT iajs-2436 108 33 must must AUX iajs-2436 108 34 equal equal VERB iajs-2436 108 35 to to ADP iajs-2436 108 36 2 2 NUM iajs-2436 108 37 when when SCONJ iajs-2436 108 38 n n X iajs-2436 108 39 is be AUX iajs-2436 108 40 even even ADV iajs-2436 108 41 and and CCONJ iajs-2436 108 42 equal equal ADJ iajs-2436 108 43 to to ADP iajs-2436 108 44 3 3 NUM iajs-2436 108 45 when when SCONJ iajs-2436 108 46 n n PRON iajs-2436 108 47 is be AUX iajs-2436 108 48 odd odd ADJ iajs-2436 108 49 , , PUNCT iajs-2436 108 50 and and CCONJ iajs-2436 108 51 the the DET iajs-2436 108 52 vertex vertex NOUN iajs-2436 108 53 0 0 PUNCT iajs-2436 108 54 adjacent adjacent ADJ iajs-2436 108 55 to to ADP iajs-2436 108 56 all all DET iajs-2436 108 57 vertices vertex NOUN iajs-2436 108 58 , , PUNCT iajs-2436 108 59 implies imply VERB iajs-2436 108 60 that that SCONJ iajs-2436 109 1 𝒳 𝒳 PROPN iajs-2436 109 2 ( ( PUNCT iajs-2436 109 3 p p NOUN iajs-2436 109 4 - - PUNCT iajs-2436 109 5 vg(r vg(r NOUN iajs-2436 109 6 ) ) PUNCT iajs-2436 109 7 ) ) PUNCT iajs-2436 109 8 equal equal ADJ iajs-2436 109 9 to to ADP iajs-2436 109 10 3 3 NUM iajs-2436 109 11 if if SCONJ iajs-2436 109 12 n n NOUN iajs-2436 109 13 is be AUX iajs-2436 109 14 even even ADV iajs-2436 109 15 and and CCONJ iajs-2436 109 16 equal equal ADJ iajs-2436 109 17 to to ADP iajs-2436 109 18 4 4 NUM iajs-2436 109 19 if if SCONJ iajs-2436 109 20 n n ADJ iajs-2436 109 21 is be AUX iajs-2436 109 22 odd odd ADJ iajs-2436 109 23 , , PUNCT iajs-2436 109 24 hence hence ADV iajs-2436 109 25 𝒳 𝒳 PROPN iajs-2436 109 26 ( ( PUNCT iajs-2436 109 27 p p NOUN iajs-2436 109 28 - - PUNCT iajs-2436 109 29 vg(r vg(r NOUN iajs-2436 109 30 ) ) PUNCT iajs-2436 109 31 ) ) PUNCT iajs-2436 110 1 = = SYM iajs-2436 110 2 𝒳 𝒳 PROPN iajs-2436 110 3 ( ( PUNCT iajs-2436 110 4 g(c g(c VERB iajs-2436 110 5 [ [ X iajs-2436 110 6 r r X iajs-2436 110 7 ] ] PUNCT iajs-2436 110 8 ) ) PUNCT iajs-2436 111 1 +1 +1 PROPN iajs-2436 111 2 . . PUNCT iajs-2436 111 3 case case NOUN iajs-2436 111 4 iii iii X iajs-2436 111 5 : : PUNCT iajs-2436 111 6 let let VERB iajs-2436 111 7 g(c g(c VERB iajs-2436 111 8 [ [ X iajs-2436 111 9 r r X iajs-2436 111 10 ] ] X iajs-2436 111 11 is be AUX iajs-2436 111 12 an an DET iajs-2436 111 13 i i NOUN iajs-2436 111 14 - - PUNCT iajs-2436 111 15 partite partite ADJ iajs-2436 111 16 graph graph NOUN iajs-2436 111 17 where where SCONJ iajs-2436 111 18 g(c g(c VERB iajs-2436 111 19 [ [ X iajs-2436 111 20 r r X iajs-2436 111 21 ] ] X iajs-2436 111 22 ) ) PUNCT iajs-2436 111 23 = = SYM iajs-2436 111 24 𝐾 𝐾 PROPN iajs-2436 111 25 , , PUNCT iajs-2436 111 26 , , PUNCT iajs-2436 111 27 … … PUNCT iajs-2436 111 28 , , PUNCT iajs-2436 111 29 , , PUNCT iajs-2436 111 30 then then ADV iajs-2436 111 31 𝒳 𝒳 PROPN iajs-2436 111 32 ( ( PUNCT iajs-2436 111 33 g(c g(c VERB iajs-2436 111 34 [ [ X iajs-2436 111 35 r r X iajs-2436 111 36 ] ] X iajs-2436 111 37 ) ) PUNCT iajs-2436 111 38 ) ) PUNCT iajs-2436 112 1 = = PUNCT iajs-2436 113 1 i i PRON iajs-2436 113 2 , , PUNCT iajs-2436 113 3 then then ADV iajs-2436 113 4 we we PRON iajs-2436 113 5 have have VERB iajs-2436 113 6 𝒳 𝒳 PROPN iajs-2436 113 7 ( ( PUNCT iajs-2436 113 8 p p NOUN iajs-2436 113 9 - - PUNCT iajs-2436 113 10 vg(r vg(r NOUN iajs-2436 113 11 ) ) PUNCT iajs-2436 113 12 ) ) PUNCT iajs-2436 114 1 = = PUNCT iajs-2436 115 1 i+1 i+1 NOUN iajs-2436 115 2 = = SYM iajs-2436 115 3 𝒳 𝒳 PROPN iajs-2436 115 4 ( ( PUNCT iajs-2436 115 5 g(c g(c VERB iajs-2436 115 6 [ [ X iajs-2436 115 7 r r X iajs-2436 115 8 ] ] PUNCT iajs-2436 115 9 ) ) PUNCT iajs-2436 115 10 +1 +1 PROPN iajs-2436 115 11 case case NOUN iajs-2436 115 12 iv iv X iajs-2436 115 13 : : PUNCT iajs-2436 115 14 if if SCONJ iajs-2436 115 15 g(c g(c PROPN iajs-2436 115 16 [ [ X iajs-2436 115 17 r r NOUN iajs-2436 115 18 ] ] X iajs-2436 115 19 ) ) PUNCT iajs-2436 115 20 = = SYM iajs-2436 116 1 𝐾 𝐾 PROPN iajs-2436 116 2 then then ADV iajs-2436 116 3 the the DET iajs-2436 116 4 chromatic chromatic ADJ iajs-2436 116 5 number number NOUN iajs-2436 116 6 of of ADP iajs-2436 116 7 g(c g(c PROPN iajs-2436 116 8 [ [ X iajs-2436 116 9 r r X iajs-2436 116 10 ] ] X iajs-2436 116 11 ) ) PUNCT iajs-2436 116 12 ) ) PUNCT iajs-2436 116 13 equal equal ADJ iajs-2436 116 14 to to ADP iajs-2436 116 15 h h NOUN iajs-2436 116 16 and and CCONJ iajs-2436 116 17 therefore therefore ADV iajs-2436 116 18 𝒳 𝒳 PROPN iajs-2436 116 19 ( ( PUNCT iajs-2436 116 20 p p PROPN iajs-2436 116 21 - - PUNCT iajs-2436 116 22 vg(r vg(r NOUN iajs-2436 116 23 ) ) PUNCT iajs-2436 116 24 ) ) PUNCT iajs-2436 117 1 = = PUNCT iajs-2436 117 2 h+1 h+1 PUNCT iajs-2436 117 3 = = SYM iajs-2436 117 4 𝒳 𝒳 PROPN iajs-2436 117 5 ( ( PUNCT iajs-2436 117 6 g(c g(c VERB iajs-2436 117 7 [ [ X iajs-2436 117 8 r r X iajs-2436 117 9 ] ] PUNCT iajs-2436 117 10 ) ) PUNCT iajs-2436 118 1 +1 +1 PROPN iajs-2436 118 2 case case NOUN iajs-2436 118 3 v v NOUN iajs-2436 118 4 : : PUNCT iajs-2436 118 5 now now ADV iajs-2436 118 6 if if SCONJ iajs-2436 118 7 g(c g(c VERB iajs-2436 118 8 [ [ X iajs-2436 118 9 r r NOUN iajs-2436 118 10 ] ] PUNCT iajs-2436 118 11 ) ) PUNCT iajs-2436 118 12 is be AUX iajs-2436 118 13 a a DET iajs-2436 118 14 connected connected ADJ iajs-2436 118 15 graph graph NOUN iajs-2436 118 16 and and CCONJ iajs-2436 118 17 it it PRON iajs-2436 118 18 includes include VERB iajs-2436 118 19 a a DET iajs-2436 118 20 maximal maximal ADJ iajs-2436 118 21 clique clique NOUN iajs-2436 118 22 𝐾 𝐾 PROPN iajs-2436 118 23 , , PUNCT iajs-2436 118 24 𝑖 𝑖 ADP iajs-2436 118 25 2 2 NUM iajs-2436 118 26 then then ADV iajs-2436 118 27 the the DET iajs-2436 118 28 chromatic chromatic ADJ iajs-2436 118 29 number number NOUN iajs-2436 118 30 of of ADP iajs-2436 118 31 𝐾 𝐾 PROPN iajs-2436 118 32 equal equal ADJ iajs-2436 118 33 to to ADP iajs-2436 118 34 i i PRON iajs-2436 118 35 , , PUNCT iajs-2436 118 36 and and CCONJ iajs-2436 118 37 the the DET iajs-2436 118 38 rest rest NOUN iajs-2436 118 39 of of ADP iajs-2436 118 40 the the DET iajs-2436 118 41 vertices vertex NOUN iajs-2436 118 42 of of ADP iajs-2436 118 43 g(c g(c NOUN iajs-2436 118 44 [ [ X iajs-2436 118 45 r r X iajs-2436 118 46 ] ] X iajs-2436 118 47 ) ) PUNCT iajs-2436 118 48 can can AUX iajs-2436 118 49 be be AUX iajs-2436 118 50 colored color VERB iajs-2436 118 51 by by ADP iajs-2436 118 52 any any DET iajs-2436 118 53 colors color NOUN iajs-2436 118 54 of of ADP iajs-2436 118 55 the the DET iajs-2436 118 56 vertices vertex NOUN iajs-2436 118 57 of of ADP iajs-2436 118 58 𝐾 𝐾 PROPN iajs-2436 118 59 because because SCONJ iajs-2436 118 60 they they PRON iajs-2436 118 61 are be AUX iajs-2436 118 62 not not PART iajs-2436 118 63 associated associate VERB iajs-2436 118 64 with with ADP iajs-2436 118 65 it it PRON iajs-2436 118 66 . . PUNCT iajs-2436 119 1 then then ADV iajs-2436 119 2 𝒳 𝒳 PROPN iajs-2436 119 3 ( ( PUNCT iajs-2436 119 4 g(c g(c VERB iajs-2436 119 5 [ [ X iajs-2436 119 6 r r X iajs-2436 119 7 ] ] X iajs-2436 119 8 ) ) PUNCT iajs-2436 119 9 ) ) PUNCT iajs-2436 120 1 = = PUNCT iajs-2436 120 2 i i PRON iajs-2436 120 3 and and CCONJ iajs-2436 120 4 𝒳 𝒳 PROPN iajs-2436 120 5 ( ( PUNCT iajs-2436 120 6 p p NOUN iajs-2436 120 7 - - PUNCT iajs-2436 120 8 vg(r vg(r NOUN iajs-2436 120 9 ) ) PUNCT iajs-2436 120 10 ) ) PUNCT iajs-2436 121 1 = = PUNCT iajs-2436 122 1 i+1 i+1 NOUN iajs-2436 122 2 = = SYM iajs-2436 122 3 𝒳 𝒳 PROPN iajs-2436 122 4 ( ( PUNCT iajs-2436 122 5 g(c g(c VERB iajs-2436 122 6 [ [ X iajs-2436 122 7 r r X iajs-2436 122 8 ] ] PUNCT iajs-2436 122 9 ) ) PUNCT iajs-2436 122 10 +1 +1 PROPN iajs-2436 122 11 case case NOUN iajs-2436 122 12 vi vi NOUN iajs-2436 122 13 : : PUNCT iajs-2436 122 14 let let VERB iajs-2436 122 15 𝐺 𝐺 PROPN iajs-2436 122 16 , , PUNCT iajs-2436 122 17 𝐺 𝐺 PROPN iajs-2436 122 18 , , PUNCT iajs-2436 122 19 … … PUNCT iajs-2436 122 20 , , PUNCT iajs-2436 122 21 𝐺 𝐺 PROPN iajs-2436 122 22 be be VERB iajs-2436 122 23 a a DET iajs-2436 122 24 disjoint disjoint ADJ iajs-2436 122 25 components component NOUN iajs-2436 122 26 of of ADP iajs-2436 122 27 g(c g(c PROPN iajs-2436 122 28 [ [ X iajs-2436 122 29 r r X iajs-2436 122 30 ] ] PUNCT iajs-2436 122 31 ) ) PUNCT iajs-2436 122 32 then then ADV iajs-2436 122 33 𝒳 𝒳 PROPN iajs-2436 122 34 ( ( PUNCT iajs-2436 122 35 g(c g(c VERB iajs-2436 122 36 [ [ X iajs-2436 122 37 r r X iajs-2436 122 38 ] ] X iajs-2436 122 39 ) ) PUNCT iajs-2436 122 40 ) ) PUNCT iajs-2436 123 1 = = SYM iajs-2436 123 2 max max PROPN iajs-2436 123 3 { { PUNCT iajs-2436 123 4 𝒳𝐺 𝒳𝐺 PROPN iajs-2436 123 5 , , PUNCT iajs-2436 123 6 𝒳𝐺 𝒳𝐺 PROPN iajs-2436 123 7 , , PUNCT iajs-2436 123 8 … … PUNCT iajs-2436 123 9 , , PUNCT iajs-2436 123 10 𝒳𝐺 𝒳𝐺 PROPN iajs-2436 123 11 } } PUNCT iajs-2436 123 12 , , PUNCT iajs-2436 123 13 suppose suppose VERB iajs-2436 123 14 𝒳 𝒳 PROPN iajs-2436 123 15 ( ( PUNCT iajs-2436 123 16 g(c g(c VERB iajs-2436 123 17 [ [ X iajs-2436 123 18 r r X iajs-2436 123 19 ] ] X iajs-2436 123 20 ) ) PUNCT iajs-2436 123 21 ) ) PUNCT iajs-2436 124 1 = = SYM iajs-2436 124 2 s s VERB iajs-2436 124 3 now now ADV iajs-2436 124 4 if if SCONJ iajs-2436 124 5 we we PRON iajs-2436 124 6 have have VERB iajs-2436 124 7 a a DET iajs-2436 124 8 vertices vertex NOUN iajs-2436 124 9 of of ADP iajs-2436 124 10 p p NOUN iajs-2436 124 11 - - PUNCT iajs-2436 124 12 vg(r vg(r VERB iajs-2436 124 13 ) ) PUNCT iajs-2436 124 14 with with ADP iajs-2436 124 15 degree degree NOUN iajs-2436 124 16 1 1 NUM iajs-2436 124 17 this this PRON iajs-2436 124 18 can can AUX iajs-2436 124 19 be be AUX iajs-2436 124 20 colored color VERB iajs-2436 124 21 by by ADP iajs-2436 124 22 any any DET iajs-2436 124 23 color color NOUN iajs-2436 124 24 of of ADP iajs-2436 124 25 these these DET iajs-2436 124 26 s s PART iajs-2436 124 27 colors color NOUN iajs-2436 124 28 used use VERB iajs-2436 124 29 to to PART iajs-2436 124 30 color color VERB iajs-2436 124 31 g(c g(c NOUN iajs-2436 125 1 [ [ X iajs-2436 125 2 r r X iajs-2436 125 3 ] ] X iajs-2436 125 4 ) ) PUNCT iajs-2436 125 5 , , PUNCT iajs-2436 125 6 and and CCONJ iajs-2436 125 7 the the DET iajs-2436 125 8 vertex vertex NOUN iajs-2436 125 9 0 0 PUNCT iajs-2436 125 10 has have VERB iajs-2436 125 11 another another DET iajs-2436 125 12 color color NOUN iajs-2436 125 13 since since SCONJ iajs-2436 125 14 it it PRON iajs-2436 125 15 adjacent adjacent ADJ iajs-2436 125 16 to to ADP iajs-2436 125 17 all all DET iajs-2436 125 18 vertices vertex NOUN iajs-2436 125 19 , , PUNCT iajs-2436 125 20 this this PRON iajs-2436 125 21 implies imply VERB iajs-2436 126 1 that that SCONJ iajs-2436 126 2 𝒳 𝒳 PROPN iajs-2436 126 3 ( ( PUNCT iajs-2436 126 4 p p NOUN iajs-2436 126 5 - - PUNCT iajs-2436 126 6 vg(r vg(r NOUN iajs-2436 126 7 ) ) PUNCT iajs-2436 126 8 ) ) PUNCT iajs-2436 126 9 = = PUNCT iajs-2436 126 10 s+1= s+1= VERB iajs-2436 126 11 𝒳 𝒳 PROPN iajs-2436 126 12 ( ( PUNCT iajs-2436 126 13 g(c g(c VERB iajs-2436 126 14 [ [ X iajs-2436 126 15 r r X iajs-2436 126 16 ] ] PUNCT iajs-2436 126 17 ) ) PUNCT iajs-2436 126 18 +1 +1 INTJ iajs-2436 126 19 . . PUNCT iajs-2436 127 1 from from ADP iajs-2436 127 2 all all PRON iajs-2436 127 3 above above ADP iajs-2436 127 4 cases case NOUN iajs-2436 127 5 then then ADV iajs-2436 127 6 𝒳 𝒳 PROPN iajs-2436 127 7 ( ( PUNCT iajs-2436 127 8 p p NOUN iajs-2436 127 9 - - PUNCT iajs-2436 127 10 vg(r vg(r NOUN iajs-2436 127 11 ) ) PUNCT iajs-2436 127 12 ) ) PUNCT iajs-2436 128 1 = = SYM iajs-2436 128 2 𝒳 𝒳 PROPN iajs-2436 128 3 ( ( PUNCT iajs-2436 128 4 g(c g(c VERB iajs-2436 128 5 [ [ X iajs-2436 128 6 r r X iajs-2436 128 7 ] ] PUNCT iajs-2436 128 8 ) ) PUNCT iajs-2436 129 1 +1 +1 PROPN iajs-2436 129 2 lemma lemma PROPN iajs-2436 129 3 3.6 3.6 NUM iajs-2436 129 4 let let VERB iajs-2436 129 5 r=𝑍 r=𝑍 PROPN iajs-2436 129 6 be be AUX iajs-2436 129 7 a a DET iajs-2436 129 8 ring ring NOUN iajs-2436 129 9 , , PUNCT iajs-2436 129 10 where where SCONJ iajs-2436 129 11 p p NOUN iajs-2436 129 12 3 3 NUM iajs-2436 129 13 is be AUX iajs-2436 129 14 prime prime ADJ iajs-2436 129 15 number number NOUN iajs-2436 129 16 then then ADV iajs-2436 129 17 𝒳 𝒳 PROPN iajs-2436 129 18 ( ( PUNCT iajs-2436 129 19 p p NOUN iajs-2436 129 20 - - PUNCT iajs-2436 129 21 vg(r vg(r NOUN iajs-2436 129 22 ) ) PUNCT iajs-2436 129 23 ) ) PUNCT iajs-2436 130 1 =3 =3 VERB iajs-2436 130 2 . . PUNCT iajs-2436 130 3     SPACE iajs-2436 131 1 153 153 NUM iajs-2436 131 2     SPACE iajs-2436 131 3 ibn ibn PROPN iajs-2436 131 4 al al PROPN iajs-2436 131 5 - - PUNCT iajs-2436 131 6 haitham haitham PROPN iajs-2436 131 7 jour jour X iajs-2436 131 8 . . PROPN iajs-2436 131 9 for for ADP iajs-2436 131 10 pure pure ADJ iajs-2436 131 11 & & CCONJ iajs-2436 131 12 appl appl PROPN iajs-2436 131 13 . . PUNCT iajs-2436 132 1 sci sci PROPN iajs-2436 132 2 . . PROPN iajs-2436 133 1 33 33 NUM iajs-2436 133 2 ( ( PUNCT iajs-2436 133 3 2 2 NUM iajs-2436 133 4 ) ) PUNCT iajs-2436 133 5 2020 2020 NUM iajs-2436 134 1 proof proof NOUN iajs-2436 134 2 let let VERB iajs-2436 134 3 𝑎𝑏 𝑎𝑏 PRON iajs-2436 134 4 ∈ ∈ PROPN iajs-2436 134 5 vg vg NOUN iajs-2436 134 6 ( ( PUNCT iajs-2436 134 7 r r NOUN iajs-2436 134 8 ) ) PUNCT iajs-2436 134 9 then then ADV iajs-2436 134 10 𝑎 𝑎 X iajs-2436 134 11 𝑎 𝑎 NOUN iajs-2436 134 12 𝑏 𝑏 NOUN iajs-2436 134 13 or or CCONJ iajs-2436 134 14 𝑏 𝑏 PRON iajs-2436 134 15 𝑎 𝑎 NOUN iajs-2436 134 16 , , PUNCT iajs-2436 134 17 𝑎 𝑎 NOUN iajs-2436 134 18 𝑏 𝑏 NOUN iajs-2436 134 19 and and CCONJ iajs-2436 134 20 𝑎 𝑎 ADJ iajs-2436 134 21 0 0 NUM iajs-2436 134 22 𝑏 𝑏 NOUN iajs-2436 134 23 , , PUNCT iajs-2436 134 24 since since SCONJ iajs-2436 134 25 p p NOUN iajs-2436 134 26 is be AUX iajs-2436 134 27 prime prime ADJ iajs-2436 134 28 then then ADV iajs-2436 134 29 r r NOUN iajs-2436 134 30 is be AUX iajs-2436 134 31 afield afield VERB iajs-2436 134 32 implies imply VERB iajs-2436 134 33 that that SCONJ iajs-2436 134 34 𝑏 𝑏 PROPN iajs-2436 134 35 𝑎 𝑎 NOUN iajs-2436 134 36 or or CCONJ iajs-2436 134 37 𝑎 𝑎 PRON iajs-2436 134 38 𝑏 𝑏 NOUN iajs-2436 134 39 is be AUX iajs-2436 134 40 unique unique ADJ iajs-2436 134 41 number number NOUN iajs-2436 134 42 satisfy satisfy VERB iajs-2436 134 43 the the DET iajs-2436 134 44 equation equation NOUN iajs-2436 134 45 , , PUNCT iajs-2436 134 46 since since SCONJ iajs-2436 134 47 𝑎0 𝑎0 PROPN iajs-2436 134 48 , , PUNCT iajs-2436 134 49 𝑏0 𝑏0 PROPN iajs-2436 134 50 ∈ ∈ PROPN iajs-2436 134 51 e(p e(p NOUN iajs-2436 134 52 - - PUNCT iajs-2436 134 53 vg(r vg(r VERB iajs-2436 134 54 ) ) PUNCT iajs-2436 134 55 ) ) PUNCT iajs-2436 135 1 then then ADV iajs-2436 135 2 p p X iajs-2436 135 3 - - PUNCT iajs-2436 135 4 vg(r vg(r NOUN iajs-2436 135 5 ) ) PUNCT iajs-2436 135 6 graph graph NOUN iajs-2436 135 7 has have VERB iajs-2436 135 8 cycle cycle NOUN iajs-2436 135 9 𝐶 𝐶 PROPN iajs-2436 135 10 , , PUNCT iajs-2436 135 11 then then ADV iajs-2436 135 12 by by ADP iajs-2436 135 13 ( ( PUNCT iajs-2436 135 14 throrem throrem NUM iajs-2436 135 15 2.14 2.14 NUM iajs-2436 135 16 ) ) PUNCT iajs-2436 135 17 𝒳 𝒳 PROPN iajs-2436 135 18 ( ( PUNCT iajs-2436 135 19 p p NOUN iajs-2436 135 20 - - PUNCT iajs-2436 135 21 vg(r vg(r NOUN iajs-2436 135 22 ) ) PUNCT iajs-2436 135 23 ) ) PUNCT iajs-2436 136 1 =3 =3 VERB iajs-2436 136 2 example example NOUN iajs-2436 136 3 3.7 3.7 NUM iajs-2436 136 4 the the DET iajs-2436 136 5 chromatic chromatic ADJ iajs-2436 136 6 number number NOUN iajs-2436 136 7 of of ADP iajs-2436 136 8 p p PROPN iajs-2436 136 9 – – PUNCT iajs-2436 136 10 von von PROPN iajs-2436 136 11 neumann neumann PROPN iajs-2436 136 12 regular regular ADJ iajs-2436 136 13 graph graph NOUN iajs-2436 136 14 of of ADP iajs-2436 136 15 𝑍 𝑍 PROPN iajs-2436 136 16 , , PUNCT iajs-2436 136 17 𝑍 𝑍 NOUN iajs-2436 136 18 and and CCONJ iajs-2436 136 19 𝑍 𝑍 PROPN iajs-2436 136 20 are be AUX iajs-2436 136 21 4 4 NUM iajs-2436 136 22 figure figure NOUN iajs-2436 136 23 5 5 NUM iajs-2436 136 24 :p :p NOUN iajs-2436 136 25 - - PUNCT iajs-2436 136 26 vg(z vg(z PUNCT iajs-2436 136 27 ) ) PUNCT iajs-2436 136 28 figure figure VERB iajs-2436 136 29 6 6 NUM iajs-2436 136 30 : : PUNCT iajs-2436 136 31 p p NUM iajs-2436 136 32 - - PUNCT iajs-2436 136 33 vg(𝑍 vg(𝑍 ADV iajs-2436 136 34 ) ) PUNCT iajs-2436 136 35 theorem theorem VERB iajs-2436 136 36 3.8 3.8 NUM iajs-2436 136 37 the the DET iajs-2436 136 38 ring ring NOUN iajs-2436 136 39 𝑍 𝑍 NOUN iajs-2436 136 40 , , PUNCT iajs-2436 136 41 𝒳 𝒳 PROPN iajs-2436 136 42 ( ( PUNCT iajs-2436 136 43 p p NOUN iajs-2436 136 44 - - PUNCT iajs-2436 136 45 vg(𝑍 vg(𝑍 PROPN iajs-2436 136 46 ) ) PUNCT iajs-2436 136 47 ) ) PUNCT iajs-2436 137 1 =3 =3 VERB iajs-2436 137 2 where where SCONJ iajs-2436 137 3 p p PROPN iajs-2436 137 4 3 3 NUM iajs-2436 137 5 is be AUX iajs-2436 137 6 prime prime ADJ iajs-2436 137 7 number number NOUN iajs-2436 137 8 . . PUNCT iajs-2436 138 1 example example NOUN iajs-2436 138 2 3.9 3.9 NUM iajs-2436 138 3 r= r= ADJ iajs-2436 138 4 𝑍 𝑍 NOUN iajs-2436 138 5 then then ADV iajs-2436 138 6 𝒳 𝒳 PROPN iajs-2436 138 7 ( ( PUNCT iajs-2436 138 8 p p NOUN iajs-2436 138 9 - - PUNCT iajs-2436 138 10 vg(𝑍 vg(𝑍 PRON iajs-2436 138 11 ) ) PUNCT iajs-2436 138 12 ) ) PUNCT iajs-2436 139 1 = = SYM iajs-2436 139 2 3 3 NUM iajs-2436 139 3 . . NOUN iajs-2436 140 1 6 6 NUM iajs-2436 140 2 9 9 NUM iajs-2436 140 3 4 4 NUM iajs-2436 140 4 7 7 NUM iajs-2436 140 5 8 8 NUM iajs-2436 140 6 2 2 NUM iajs-2436 140 7 3 3 NUM iajs-2436 140 8 5 5 NUM iajs-2436 140 9 1 1 NUM iajs-2436 140 10 0 0 NUM iajs-2436 140 11 4 4 NUM iajs-2436 140 12 12 12 NUM iajs-2436 140 13 2 2 NUM iajs-2436 140 14 11 11 NUM iajs-2436 140 15 9 9 NUM iajs-2436 140 16 613 613 NUM iajs-2436 140 17 7 7 NUM iajs-2436 140 18 1 1 NUM iajs-2436 140 19 8 8 NUM iajs-2436 140 20 3 3 NUM iajs-2436 140 21 5 5 NUM iajs-2436 140 22 10 10 NUM iajs-2436 140 23 0 0 NUM iajs-2436 140 24     SPACE iajs-2436 140 25 154 154 NUM iajs-2436 140 26     SPACE iajs-2436 140 27 ibn ibn PROPN iajs-2436 140 28 al al PROPN iajs-2436 140 29 - - PUNCT iajs-2436 140 30 haitham haitham PROPN iajs-2436 140 31 jour jour X iajs-2436 140 32 . . PROPN iajs-2436 140 33 for for ADP iajs-2436 140 34 pure pure ADJ iajs-2436 140 35 & & CCONJ iajs-2436 140 36 appl appl PROPN iajs-2436 140 37 . . PUNCT iajs-2436 141 1 sci sci PROPN iajs-2436 141 2 . . PROPN iajs-2436 142 1 33 33 NUM iajs-2436 142 2 ( ( PUNCT iajs-2436 142 3 2 2 NUM iajs-2436 142 4 ) ) PUNCT iajs-2436 142 5 2020 2020 NUM iajs-2436 142 6 figure figure VERB iajs-2436 142 7 7 7 NUM iajs-2436 142 8 : : PUNCT iajs-2436 142 9 p p NUM iajs-2436 142 10 - - PUNCT iajs-2436 142 11 vg(𝑍 vg(𝑍 ADV iajs-2436 142 12 ) ) PUNCT iajs-2436 142 13 note note VERB iajs-2436 142 14 3.10 3.10 NUM iajs-2436 142 15 if if SCONJ iajs-2436 142 16 r= r= ADJ iajs-2436 142 17 𝑍 𝑍 NOUN iajs-2436 142 18 and and CCONJ iajs-2436 142 19 𝑎𝑏 𝑎𝑏 ADP iajs-2436 142 20 ∈ ∈ PROPN iajs-2436 142 21 e e X iajs-2436 142 22 ( ( PUNCT iajs-2436 142 23 p p NOUN iajs-2436 142 24 - - PUNCT iajs-2436 142 25 vg(𝑍 vg(𝑍 PROPN iajs-2436 142 26 ) ) PUNCT iajs-2436 142 27 ) ) PUNCT iajs-2436 143 1 then then ADV iajs-2436 143 2 𝑎 𝑎 X iajs-2436 143 3 𝑎 𝑎 NOUN iajs-2436 143 4 𝑏 𝑏 NOUN iajs-2436 143 5 and and CCONJ iajs-2436 143 6 𝑏 𝑏 NOUN iajs-2436 143 7 𝑏 𝑏 PROPN iajs-2436 143 8 𝑎 𝑎 PROPN iajs-2436 143 9 and and CCONJ iajs-2436 143 10 ab ab PROPN iajs-2436 143 11 mod mod PROPN iajs-2436 144 1 p p PROPN iajs-2436 144 2 = = PROPN iajs-2436 144 3 1 1 NUM iajs-2436 144 4 . . PUNCT iajs-2436 144 5 theorem theorem VERB iajs-2436 144 6 3.11 3.11 NUM iajs-2436 144 7 the the DET iajs-2436 144 8 ring ring NOUN iajs-2436 144 9 𝑍 𝑍 NOUN iajs-2436 144 10 , , PUNCT iajs-2436 144 11 𝒳 𝒳 PROPN iajs-2436 144 12 ( ( PUNCT iajs-2436 144 13 p p NOUN iajs-2436 144 14 - - PUNCT iajs-2436 144 15 vg(𝑍 vg(𝑍 NOUN iajs-2436 144 16 ) ) PUNCT iajs-2436 144 17 ) ) PUNCT iajs-2436 145 1 =3 =3 VERB iajs-2436 145 2 , , PUNCT iajs-2436 145 3 when when SCONJ iajs-2436 145 4 p p NOUN iajs-2436 145 5 3 3 NUM iajs-2436 145 6 is be AUX iajs-2436 145 7 a a DET iajs-2436 145 8 prime prime ADJ iajs-2436 145 9 number number NOUN iajs-2436 145 10 and and CCONJ iajs-2436 145 11 n n DET iajs-2436 145 12 a a DET iajs-2436 145 13 positive positive ADJ iajs-2436 145 14 integer integer NOUN iajs-2436 145 15 . . PUNCT iajs-2436 146 1 proof proof NOUN iajs-2436 146 2 let let VERB iajs-2436 146 3 a a DET iajs-2436 146 4 , , PUNCT iajs-2436 146 5 b b X iajs-2436 146 6 ∈ ∈ PROPN iajs-2436 146 7 𝑍 𝑍 NOUN iajs-2436 146 8 and and CCONJ iajs-2436 146 9 a a DET iajs-2436 146 10 b b NOUN iajs-2436 146 11 , , PUNCT iajs-2436 146 12 a a DET iajs-2436 146 13 0 0 NUM iajs-2436 146 14 b b NOUN iajs-2436 146 15 and and CCONJ iajs-2436 146 16 𝑎𝑏 𝑎𝑏 PROPN iajs-2436 146 17 ∈ ∈ PROPN iajs-2436 146 18 e(p e(p NOUN iajs-2436 146 19 - - PUNCT iajs-2436 146 20 vg(𝑍 vg(𝑍 PROPN iajs-2436 146 21 ) ) PUNCT iajs-2436 146 22 ) ) PUNCT iajs-2436 147 1 then then ADV iajs-2436 147 2 𝑎 𝑎 X iajs-2436 147 3 𝑎 𝑎 NOUN iajs-2436 147 4 𝑏 𝑏 NOUN iajs-2436 147 5 and and CCONJ iajs-2436 147 6 𝑏 𝑏 NOUN iajs-2436 147 7 𝑏 𝑏 PROPN iajs-2436 147 8 𝑎 𝑎 NOUN iajs-2436 147 9 and and CCONJ iajs-2436 147 10 ab=1 ab=1 PROPN iajs-2436 147 11 and and CCONJ iajs-2436 147 12 since since SCONJ iajs-2436 147 13 b b NOUN iajs-2436 147 14 is be AUX iajs-2436 147 15 unique unique ADJ iajs-2436 147 16 ( ( PUNCT iajs-2436 147 17 because because SCONJ iajs-2436 147 18 b b X iajs-2436 147 19 is be AUX iajs-2436 147 20 the the DET iajs-2436 147 21 inverse inverse NOUN iajs-2436 147 22 of of ADP iajs-2436 147 23 a a PRON iajs-2436 147 24 , , PUNCT iajs-2436 147 25 and and CCONJ iajs-2436 147 26 it it PRON iajs-2436 147 27 is be AUX iajs-2436 147 28 only only ADV iajs-2436 147 29 element element NOUN iajs-2436 147 30 satisfy satisfy VERB iajs-2436 147 31 the the DET iajs-2436 147 32 equation equation NOUN iajs-2436 147 33 ) ) PUNCT iajs-2436 147 34 then then ADV iajs-2436 147 35 p p X iajs-2436 147 36 - - PUNCT iajs-2436 147 37 vg(𝑍 vg(𝑍 PRON iajs-2436 147 38 ) ) PUNCT iajs-2436 147 39 has have VERB iajs-2436 147 40 only only ADV iajs-2436 147 41 cycle cycle NOUN iajs-2436 147 42 𝐶 𝐶 PROPN iajs-2436 147 43 by by ADP iajs-2436 147 44 theorem theorem NOUN iajs-2436 147 45 ( ( PUNCT iajs-2436 147 46 2.14 2.14 NUM iajs-2436 147 47 ) ) PUNCT iajs-2436 147 48 every every DET iajs-2436 147 49 cycle cycle NOUN iajs-2436 147 50 has have VERB iajs-2436 147 51 odd odd ADJ iajs-2436 147 52 vertices vertex NOUN iajs-2436 147 53 has have VERB iajs-2436 147 54 chromatic chromatic ADJ iajs-2436 147 55 number number NOUN iajs-2436 147 56 equal equal ADJ iajs-2436 147 57 to to ADP iajs-2436 147 58 3 3 NUM iajs-2436 147 59 and and CCONJ iajs-2436 147 60 in in ADP iajs-2436 147 61 any any DET iajs-2436 147 62 p p NOUN iajs-2436 147 63 - - PUNCT iajs-2436 147 64 vg(𝑍 vg(𝑍 NOUN iajs-2436 147 65 ) ) PUNCT iajs-2436 147 66 graph graph NOUN iajs-2436 147 67 has have VERB iajs-2436 147 68 more more ADJ iajs-2436 147 69 than than ADP iajs-2436 147 70 one one NUM iajs-2436 147 71 𝐶 𝐶 PROPN iajs-2436 147 72 , , PUNCT iajs-2436 147 73 and and CCONJ iajs-2436 147 74 all all DET iajs-2436 147 75 𝐶 𝐶 PROPN iajs-2436 147 76 in in ADP iajs-2436 147 77 a a DET iajs-2436 147 78 graph graph NOUN iajs-2436 147 79 commend commend NOUN iajs-2436 147 80 in in ADP iajs-2436 147 81 one one NUM iajs-2436 147 82 vertex vertex NOUN iajs-2436 147 83 ( ( PUNCT iajs-2436 147 84 0 0 NUM iajs-2436 147 85 ) ) PUNCT iajs-2436 147 86 , , PUNCT iajs-2436 147 87 so so CCONJ iajs-2436 147 88 the the DET iajs-2436 147 89 vertex vertex NOUN iajs-2436 147 90 has have VERB iajs-2436 147 91 one one NUM iajs-2436 147 92 color color NOUN iajs-2436 147 93 it it PRON iajs-2436 147 94 is be AUX iajs-2436 147 95 clear clear ADJ iajs-2436 147 96 that that SCONJ iajs-2436 147 97 the the DET iajs-2436 147 98 color color NOUN iajs-2436 147 99 of of ADP iajs-2436 147 100 other other ADJ iajs-2436 147 101 vertices vertex NOUN iajs-2436 147 102 in in ADP iajs-2436 147 103 the the DET iajs-2436 147 104 same same ADJ iajs-2436 147 105 cycle cycle NOUN iajs-2436 147 106 are be AUX iajs-2436 147 107 two two NUM iajs-2436 147 108 another another DET iajs-2436 147 109 colors color NOUN iajs-2436 147 110 , , PUNCT iajs-2436 147 111 so so ADV iajs-2436 147 112 we we PRON iajs-2436 147 113 have have VERB iajs-2436 147 114 only only ADV iajs-2436 147 115 three three NUM iajs-2436 147 116 colors color NOUN iajs-2436 147 117 in in ADP iajs-2436 147 118 any any DET iajs-2436 147 119 p p NOUN iajs-2436 147 120 - - PUNCT iajs-2436 147 121 vg(𝑍 vg(𝑍 PROPN iajs-2436 147 122 ) ) PUNCT iajs-2436 147 123 . . PUNCT iajs-2436 148 1 4 4 X iajs-2436 148 2 . . X iajs-2436 148 3 conclusion conclusion NOUN iajs-2436 148 4 in in ADP iajs-2436 148 5 this this DET iajs-2436 148 6 work work NOUN iajs-2436 148 7 we we PRON iajs-2436 148 8 gave give VERB iajs-2436 148 9 a a DET iajs-2436 148 10 definition definition NOUN iajs-2436 148 11 of of ADP iajs-2436 148 12 pseudo pseudo NOUN iajs-2436 148 13 - - PROPN iajs-2436 148 14 von von PROPN iajs-2436 148 15 neumann neumann PROPN iajs-2436 148 16 regular regular ADJ iajs-2436 148 17 graph graph NOUN iajs-2436 148 18 p p NOUN iajs-2436 148 19 - - PUNCT iajs-2436 148 20 vg(r vg(r NOUN iajs-2436 148 21 ) ) PUNCT iajs-2436 148 22 then then ADV iajs-2436 148 23 proved prove VERB iajs-2436 148 24 the the DET iajs-2436 148 25 chromatic chromatic ADJ iajs-2436 148 26 number number NOUN iajs-2436 148 27 𝒳 𝒳 PROPN iajs-2436 148 28 ( ( PUNCT iajs-2436 148 29 p p NOUN iajs-2436 148 30 - - PUNCT iajs-2436 148 31 vg(r vg(r NOUN iajs-2436 148 32 ) ) PUNCT iajs-2436 148 33 ) ) PUNCT iajs-2436 149 1 = = SYM iajs-2436 149 2 𝒳 𝒳 PROPN iajs-2436 149 3 ( ( PUNCT iajs-2436 149 4 g(c g(c VERB iajs-2436 149 5 [ [ X iajs-2436 149 6 r r X iajs-2436 149 7 ] ] PUNCT iajs-2436 149 8 ) ) PUNCT iajs-2436 150 1 + + CCONJ iajs-2436 150 2 1 1 NUM iajs-2436 150 3 when when SCONJ iajs-2436 150 4 𝑎 𝑎 DET iajs-2436 150 5 𝑏 𝑏 NOUN iajs-2436 150 6 0 0 NUM iajs-2436 150 7 and and CCONJ iajs-2436 150 8 𝒳 𝒳 PROPN iajs-2436 150 9 ( ( PUNCT iajs-2436 150 10 p p NOUN iajs-2436 150 11 - - PUNCT iajs-2436 150 12 vg(𝑍 vg(𝑍 PROPN iajs-2436 150 13 ) ) PUNCT iajs-2436 150 14 ) ) PUNCT iajs-2436 150 15 , , PUNCT iajs-2436 150 16 𝑝 𝑝 NOUN iajs-2436 150 17 3 3 NUM iajs-2436 150 18 is be AUX iajs-2436 150 19 equal equal ADJ iajs-2436 150 20 to to ADP iajs-2436 150 21 3 3 NUM iajs-2436 150 22 , , PUNCT iajs-2436 150 23 and and CCONJ iajs-2436 150 24 𝒳 𝒳 PROPN iajs-2436 150 25 ( ( PUNCT iajs-2436 150 26 p p NOUN iajs-2436 150 27 - - PUNCT iajs-2436 150 28 vg(𝑍 vg(𝑍 PROPN iajs-2436 150 29 ) ) PUNCT iajs-2436 150 30 ) ) PUNCT iajs-2436 150 31 , , PUNCT iajs-2436 150 32 𝑝 𝑝 PROPN iajs-2436 150 33 3 3 NUM iajs-2436 150 34 equal equal ADJ iajs-2436 150 35 to to ADP iajs-2436 150 36 3 3 NUM iajs-2436 150 37 . . NOUN iajs-2436 150 38 2 2 NUM iajs-2436 150 39 20 20 NUM iajs-2436 150 40 15 15 NUM iajs-2436 150 41 12 12 NUM iajs-2436 150 42 2311 2311 NUM iajs-2436 150 43 9 9 NUM iajs-2436 150 44 22 22 NUM iajs-2436 150 45 8 8 NUM iajs-2436 150 46 21 21 NUM iajs-2436 150 47 5 5 NUM iajs-2436 150 48 17 17 NUM iajs-2436 150 49 0 0 NUM iajs-2436 150 50 14 14 NUM iajs-2436 150 51 24 24 NUM iajs-2436 150 52 18 18 NUM iajs-2436 150 53 7 7 NUM iajs-2436 150 54 6 6 NUM iajs-2436 150 55 19 19 NUM iajs-2436 150 56 4 4 NUM iajs-2436 150 57 3 3 NUM iajs-2436 150 58 13 13 NUM iajs-2436 150 59 10 10 NUM iajs-2436 150 60 16 16 NUM iajs-2436 150 61 1 1 NUM iajs-2436 150 62 0 0 NUM iajs-2436 150 63     SPACE iajs-2436 150 64 155 155 NUM iajs-2436 150 65     SPACE iajs-2436 150 66 ibn ibn PROPN iajs-2436 150 67 al al PROPN iajs-2436 150 68 - - PUNCT iajs-2436 150 69 haitham haitham PROPN iajs-2436 150 70 jour jour X iajs-2436 150 71 . . PROPN iajs-2436 151 1 for for ADP iajs-2436 151 2 pure pure ADJ iajs-2436 151 3 & & CCONJ iajs-2436 151 4 appl appl PROPN iajs-2436 151 5 . . PUNCT iajs-2436 152 1 sci sci PROPN iajs-2436 152 2 . . PROPN iajs-2436 153 1 33 33 NUM iajs-2436 153 2 ( ( PUNCT iajs-2436 153 3 2 2 NUM iajs-2436 153 4 ) ) PUNCT iajs-2436 153 5 2020 2020 NUM iajs-2436 153 6 references reference NOUN iajs-2436 153 7 1 1 NUM iajs-2436 153 8 . . X iajs-2436 153 9 beck beck PROPN iajs-2436 153 10 , , PUNCT iajs-2436 153 11 i. i. PROPN iajs-2436 153 12 coloring coloring PROPN iajs-2436 153 13 of of ADP iajs-2436 153 14 commutative commutative ADJ iajs-2436 153 15 rings ring NOUN iajs-2436 153 16 , , PUNCT iajs-2436 153 17 j. j. 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