Steinberg s Conjecture is false

Transcription

1 Stinrg s Conjtur is als arxiv: v2 [math.co] 19 Apr 2016 Vinnt Cohn-Aa Mihal Hig Danil Král Zhntao Li Estan Salgao Astrat Stinrg onjtur in 1976 that vry planar graph with no yls o lngth our or iv is 3-oloral. W isprov this onjtur. 1 Introution On o major opn prolms on olorings o planar graphs is Stinrg s Conjtur rom Th onjtur assrts that vry planar graph with no yls o lngth our or iv is 3-oloral. This prolm has n attrating a sustantial amount o attntion among graph thorists, s [4, Stion 7]. It is also on o th six graph thory prolms rank with th 4-star (highst) importan in th Opn Prolm Garn [1]. As a possil approah towars proving th onjtur, Erős in 1991 (s [12]) suggst to trmin th smallst k suh that vry planar graph with no yls o lngth 4,...,k is 3-oloral. Boroin t al. [8], improving up on [2,11], show that k 7, i.., vry planar graph with no yls o lngth our, iv, six or svn is 3-oloral. Many othr rlaxations o th onjtur hav n stalish,.g. [3,6,9,14]. On th othr han, th onjtur isknown toals inthlist oloringstting [13]. Wrr thrar or urthr rsults on th onjtur an othr prolms rlat to olorings o planar graphs to [4, 10]. Départmnt Inormatiqu, UMR CNRS 8548, Éol Normal Supériur, Paris, Fran. {vohn,zhntao}@i.ns.r. Dpartmnt o Computr Sin, Univrsity o Warwik, Covntry CV4 7AL, UK. m.hig@warwik.a.uk. Th work o this author was support y th Lvrhulm Trust 2014 Philip Lvrhulm Priz o th thir author. Mathmatis Institut, DIMAP an Dpartmnt o Computr Sin, Univrsity o Warwik, Covntry CV4 7AL, UK. .kral@warwik.a.uk. Th work o this author was support y th Lvrhulm Trust 2014 Philip Lvrhulm Priz. Éol Polythniqu, Paris, Fran, an Pontiiia Univrsia Católia Chil, Santiago, Chil. asalgaovalnzula@gmail.om. This rsarhwas on uring th stay o th son, thir an ith o th authors at Éol Normal Supériur in Paris. 1

2 a g k n m o l h j i Figur 1: Th graph G 1 analyz in Lmma 1. W isprov Stinrg s Conjtur y onstruting a planar graph with no yls o lngth our or iv that is not 3-oloral. This also isprovs Strong Boraux Conjtur rom [9] an Novosiirsk 3-Color Conjtur rom [5]. Conjtur 1 (Strong Boraux Conjtur). Evry planar graph with no pair o yls o lngth thr sharing an g an with no yl o lngth iv is 3- oloral. Conjtur 2 (Novosiirsk 3-Color Conjtur). Evry planar graph without a yl o lngth thr sharing an g with a yl o lngth thr or iv is 3- oloral. W woul lik to rmark that w hav not tri to minimiz th numr o vrtis in our onstrution; w ar atually awar o th xistn o smallr ountrxampls (th smallst on with 85 vrtis) ut w i to prsnt th on that is th simplst to analyz among thos that w hav oun. 2 Countrxampl Our onstrution has thr stps. In ah o ths stps, w onstrut a partiular planar graph with no yls o lngth our or iv that has som aitional proprtis. Th nxt two lmmas prsnt th irst two stps o th onstrution. Lmma 1. Th graph G 1 that is pit in Figur 1 has no yls o lngth our or iv an thr is no 3-oloring that woul assign th sam olor to all th thr vrtis a, an. In aition, th istan twn a an is thr, twn a an is thr, an twn an is our. 2

3 a a G 1 G 1 G 1 Figur 2: Th graph G 2 analyz in Lmma 2. Th atual graph is rawn on th lt; its astrat strutur using th graph G 1 is givn on th right. Th thr ontat vrtis a, an ar rawn with ol irls. Proo. It is asy to hk that G 1 ontains no yl o lngth our or iv, th istan twn a an is thr, twn a an is thr, an twn an is our. Suppos that G 1 has a 3-oloring suh that all th thr vrtis a, an hav th sam olor, say γ. Th othr two olors must us in an altrnating way to olor th vrtis,,, g, h an i. This implis that th vrtis j, k an l hav th olor γ. Hn, th two olors irnt rom γ must us to olor th vrtis m, n an o, whih is impossil sin ths thr vrtis orm a triangl. Lmma 2. Th graph G 2 that is pit in Figur 2 has no yls o lngth our or iv an thr is no 3-oloring that woul assign th sam olor to all th thr vrtis a, an. In aition, th istan twn any pairs o th vrtis a, an is our. Proo. Not that th graph G 2 onsists o thr opis o th graph G 1 past togthr as pit in Figur 2. W start y hking that th istan twn th vrtis a an is our (th ass o th othr two pairs ar symmtri). Sin a an ar join y a path o lngth our, thir istan is at most our. Lmma 1 implis that thr is no path o lngth lss than our insi th opy o G 1 ontaining a an. On th othr han, any path passing through mor than on opy o G 1 must hav lngth at last thr insi ah travrs opy o G 1, so its total lngth annot lss than six. W onlu that th istan twn a an is our. W nxt hk that G 2 has no yl o lngth our or iv. By Lmma 1, no yl insi on o th opis o G 1 has lngth our or iv. Sin th istan 3

4 G 2 G 2 G 2 a G 2 Figur 3: Th graph G onstrut in th proo o Thorm 3. Th gray aras rprsnt th graph rom Figur 2. twn any two o th ontat vrtis o th opis o G 1 (th vrtis us to glu th opis togthr) is at last thr y Lmma 1, th lngth o any yl passing through two or thr o th opis is at last six. Th only yl that os not ontain an g insi a opy o G 1 is th yl an its lngth is thr. Thror, G 2 has no yl o lngth our or iv. It rmains to hk that G 2 has no 3-oloring suh that th vrtis a, an gt th sam olor. Suppos that G 2 has a 3-oloring suh that th vrtis a, an riv th sam olor, say γ. By Lmma 1, th olor o any o th vrtis, an is irnt rom γ. Howvr, this is impossil sin th vrtis, an orm a triangl an thr ar only thr olors in total. Hn, G 2 has no 3-oloring suh that th vrtis a, an woul gt th sam olor. W ar now ray to provi th ountrxampl onstrution. Thorm 3. Thr xists a planar graph with no yls o lngth our or iv that is not 3-oloral. Proo. Consir th graph G otain y taking our opis o th graph G 2 rom Lmma 2 an pasting thm togthr with aitional vrtis, an in th way pit in Figur 3. W irst argu that G has no yl o lngth our or iv. By Lmma 2, thr is no yl o lngth our or iv insi a singl opy o G 2. Any yl ontaining an g insi a opy o G 2 that is not ully ontain insi th opy must ontain at last our gs insi th onsir opy o G 2, sin th istan twn any two o th ontat vrtis o G 2 is our y Lmma 2. Sin suh a yl must ontain at last two gs outsi th opy o G 2, its lngth must at last six. Hn, thr is no yl with lngth our or iv that ontains an g insi a opy o G 2. Osrv that thr ar xatly 4

5 thr yls not ontaining an g insi any opy o G 2 an ths ar triangls, an. Thror, G has no yl o lngth our or iv. Suppos that G has a 3-oloring an lt γ th olor assign to th vrtx a. Sin th vrtis o th triangl must assign all thr olors an th olor o is not γ, ithr or is olor with γ. By symmtry, w an assum that th olor o is γ. Sin th vrtis a, an ar ontat vrtis o a opy o G 2, th olor o is not γ y Lmma 2. Similarly, sin th vrtis a, an ar ontat vrtis o a opy o G 2, th olor o is also not γ. Hn, non o th vrtis, an (not th vrtx is ajant to a) has th olor γ, whih is impossil sin th vrtis, an orm a triangl. W onlu that th graph G has no 3-oloring. Aknowlgmnt Th authors woul lik to thank Zněk Dvořák or his ommnts an insights on a omputr-assist approah towars a possil solution o Stinrg s Conjtur. Rrns [1] Opn Prolm Garn, [2] H. L. Aott, B. Zhou: On small as in 4-ritial graphs, Ars Comin. 32 (1991), [3] O. V. Boroin: Strutural proprtis o plan graphs without ajant triangls an an appliation to 3-olorings, J. Graph Thory 21 (1996), [4] O. V. Boroin: Colorings o plan graphs: A survy, Disrt Math. 313 (2013), [5] O. V. Boroin, A. N. Glov, T. R. Jnsn, an A. Raspau: Planar graphs without triangls ajant to yls o lngth rom 3 to 9 ar 3-oloral, Si. Elktron. Mat. Izv. 3 (2006), [6] O. V. Boroin, A. N. Glov, M. Montassir, an A. Raspau: Planar graphs without 5- an 7-yls an without ajant triangls ar 3-oloral, J. Comin. Thory Sr. B 99 (2009), [7] O. V. Boroin, A. N. Glov, an A. Raspau: Planar graphs without triangls ajant to yls o lngth rom 4 to 7 ar 3-oloral, Disrt Math. 310 (2010),

6 [8] O. V. Boroin, A. N. Glov, A. Raspau, an M. R. Salavatipour: Planar graphs without yls o lngth rom 4 to 7 ar 3-oloral, J. Comin. Thory Sr. B 93 (2005), [9] O. V. Boroin an A. Raspau: A suiint onition or planar graphs to 3-oloral, J. Comin. Thory Sr. B 88 (2003), [10] T. R. Jnsn, B. Tot: Graph oloring prolms, John Wily & Sons, [11] D. P. Sanrs, Y. Zhao: A not on th thr olor prolm, Graphs Comin. 11 (1995), [12] R. Stinrg: Th stat o th thr olor prolm, quo vais, graph thory?, Ann. Disrt Math. 55 (1993), [13] M. Voigt: A non-3-hoosal planar graph without yls o lngth 4 an 5, Disrt Math. 307 (2007), [14] B. Xu: On 3-oloral plan graphs without 5- an 7-yls, J. Comin. Thory Sr. B 96 (2006),