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A non-3-choosable planar graph without cycles of length 4 and 5

M. Voigt
2007 Discrete Mathematics  
Steinberg's question from 1975 whether every planar graph without 4-and 5-cycles is 3-colorable is still open.  ...  In this paper the analogous question for 3-choosability of such graphs is answered to the negative.  ...  There are planar graphs without cycles of length four and five which are not 3-choosable. Proof. Consider the graph of Fig. 1 .  ... 
doi:10.1016/j.disc.2005.11.041 fatcat:r6xnyme46bh4fik7smml6uvrzy

The proof of Steinberg's three coloring conjecture [article]

I. Cahit
2006 arXiv   pre-print
The well-known Steinberg's conjecture asserts that any planar graph without 4- and 5-cycles is 3 colorable.  ...  In this note we have given a short algorithmic proof of this conjecture based on the spiral chains of planar graphs proposed in the proof of the four color theorem by the author in 2004.  ...  Heckman for his interest and views on this problem.  ... 
arXiv:math/0607497v1 fatcat:5nplm5exifcbjdzudt6lthbnsa

Colorings of plane graphs without long monochromatic facial paths

Július Czap, Igor Fabrici, Stanislav Jendrol'
2020 Discussiones Mathematicae Graph Theory  
3-colorable) plane graph G t such that in any its 3-coloring (2-coloring) there is a monochromatic path of length at least t.  ...  We prove that each plane graph admits a 3-coloring (a 2-coloring) such that every monochromatic facial path has at most 3 vertices (at most 4 vertices).  ...  Every plane graph admits a 3-coloring without monochromatic facial 3-paths. Conjecture 6 holds for plane graphs without cycles of length t, for some t ∈ {3, 4, 5}.  ... 
doi:10.7151/dmgt.2319 fatcat:oplgmfzj6vgkdat6zwajurvy5m

Steinberg's Conjecture is false

Vincent Cohen-Addad, Michael Hebdige, Daniel Král', Zhentao Li, Esteban Salgado
2017 Journal of combinatorial theory. Series B (Print)  
Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. We disprove this conjecture.  ...  Acknowledgement The authors would like to thank Zdeněk Dvořák for his comments and insights on a computer-assisted approach towards a possible solution of Steinberg's Conjecture.  ...  Every planar graph without a cycle of length three sharing an edge with a cycle of length three or five is 3colorable.  ... 
doi:10.1016/j.jctb.2016.07.006 fatcat:h2unnoaqxraxlfcn2ipkc2f5ri

Steinberg's Conjecture is false [article]

Vincent Cohen-Addad, Michael Hebdige, Daniel Kral, Zhentao Li, Esteban Salgado
2016 arXiv   pre-print
Steinberg conjectured in 1976 that every planar graph with no cycles of length four or five is 3-colorable. We disprove this conjecture.  ...  Acknowledgement The authors would like to thank Zdeněk Dvořák for his comments and insights on a computer-assisted approach towards a possible solution of Steinberg's Conjecture.  ...  Every planar graph without a cycle of length three sharing an edge with a cycle of length three or five is 3colorable.  ... 
arXiv:1604.05108v2 fatcat:23r5or7ijjc2jmlvrmzk7caffm

Locally planar toroidal graphs are $5$-colorable

Michael O. Albertson, Walter R. Stromquist
1982 Proceedings of the American Mathematical Society  
Essentially this hypothesis means that small neighborhoods of the graph are planar. No similar conclusion holds for 4-colorability.  ...  If a graph can be embedded in a torus in such a way that all noncontractible cycles have length at least 8, then its vertices may be 5-colored.  ...  As the interior and boundary of a contractible 3-cycle form a planar graph any coloring of the bounding 3-cycle can be extended to a 4-coloring of the interior.  ... 
doi:10.1090/s0002-9939-1982-0640251-3 fatcat:g6m5mk4xurbwpmdk7lpokvda3a

Locally Planar Toroidal Graphs are 5-Colorable

Michael O. Albertson, Walter R. Stromquist
1982 Proceedings of the American Mathematical Society  
Essentially this hypothesis means that small neighborhoods of the graph are planar. No similar conclusion holds for 4-colorability.  ...  If a graph can be embedded in a torus in such a way that all noncontractible cycles have length at least 8, then its vertices may be 5-colored.  ...  As the interior and boundary of a contractible 3-cycle form a planar graph any coloring of the bounding 3-cycle can be extended to a 4-coloring of the interior.  ... 
doi:10.2307/2043580 fatcat:udhyykyzwzhrxf6f2vt5h4uuku

Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph [article]

Jin Xu
2012 arXiv   pre-print
The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color.  ...  and proves both the four color conjecture and the uniquely 4-colorable planar graph conjecture by mathematical method.  ...  They pointed out that there was one obvious leak on the proof of Ïf a maximal planar graph has a minimum degree of 5, it is not uniquely 4-colorable. Ḧere I also would like to thank them deeply.  ... 
arXiv:0911.1587v3 fatcat:ifjdqgpvjnh7nlitqzr26p2hcq

Planar graphs without cycles of length 4, 5, 8, or 9 are 3-choosable

Yingqian Wang, Huajing Lu, Ming Chen
2010 Discrete Mathematics  
It is shown that a planar graph without cycles of length 4, 5, 8, or 9 is 3-choosable.  ...  Note that there are planar graphs without cycles of length 4 or 5 known to be not 3-choosable, see [6, 7, 12] .  ...  Does every combination of four lengths of forbidden cycles among {4, 5, 6, 7, 8, 9} with 4 fixed ensure 3-choosability for planar graphs?  ... 
doi:10.1016/j.disc.2009.08.005 fatcat:h32fm763gjfohjgkbysnibbena

Some counterexamples associated with the three-color problem

V.A Aksionov, L.S Mel'nikov
1980 Journal of combinatorial theory. Series B (Print)  
In this paper we construct some counterexamples of non-3-colorable planar graphs, using the notion of "quasi-edges." The minimality of some quasi-edges is proved.  ...  ACKNOWLEDGMENTS We are indebted to results of this paper. V. A. Kostochka and R. Steinberg for discussion of the  ...  All known non-3-colorable planar graphs contain 4-or 5-cycles. This led R.  ... 
doi:10.1016/0095-8956(80)90051-9 fatcat:xoxmwtbuknfathgmnunckzgvi4

Planar graphs without adjacent cycles of length at most seven are 3-colorable

Oleg V. Borodin, Mickael Montassier, André Raspaud
2010 Discrete Mathematics  
We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3 ≤ i ≤ j ≤ 7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs.  ...  At the crossroad of Havel's and Steinberg's problems, Borodin and Raspaud [11] proved that every planar graph without 3-cycles at distance less than four and without 5-cycles is 3-colorable.  ...  Theorem 3 3 Every proper 3-coloring of the vertices of any face of length 8 to 11 in a connected planar graph without cycles of length 4 to 7 can be extended to a proper 3-coloring of the whole graph.  ... 
doi:10.1016/j.disc.2009.08.010 fatcat:fo2zunw57fduxgwrtsxpdfyfge

The Four Color Theorem – A New Simple Proof by Induction [article]

V. Vilfred Kamalappan
2021 arXiv   pre-print
We consider, in our proof, possible colorings of a minimum degree vertex, its adjacent vertices and adjacent vertices of these adjacent vertices of simple planar graphs.  ...  In 1976, Appel and Haken achieved a major break through by proving the four color theorem (4CT).  ...  (x) We classify graphs G into two, one with cycle (u 1 , u 2 , u 3 ) of length 3 and the other without cycle of length 3, other than triangular faces.  ... 
arXiv:1701.03511v5 fatcat:lox3xrgitfcsfg52e6dxptxwcm

Planar graphs without 4-,5- and 8-cycles are 3-colorable

Sakib A. Mondal
2011 Discussiones Mathematicae Graph Theory  
In this paper we prove that every planar graph without 4, 5 and 8-cycles is 3-colorable.  ...  [4] , where it is shown that any planar graph without cycles of length in {4, 5, 6, 7} is 3-colorable.  ...  Chen, Raspaud and Wang [8] showed that a planar graph without cycles of length 4, 6, 7 and 9 is 3-colorable.  ... 
doi:10.7151/dmgt.1579 fatcat:mkoczhcgjzdk5mjnyt42s2kl3i

Algorithms [chapter]

2011 Graph Coloring Problems  
Density of 4-Critical Planar Graphs 49 2.18 Square of Planar Graphs 50 Bibliography 51 3 Graphs on Higher Surfaces 59 3.1 Heawood's Empire Problem 59 3.2 Griinbaum's 3-Edge-Color Conjecture  ...  of Hamilton Cycles . 82 4.6 Brooks' Theorem for Triangle-Free Graphs 83 4.7 Graphs Without Large Complete Subgraphs 85 4.8 ^-Chromatic Graphs of Maximum Degree k 85 4.9 Total Coloring 86  ... 
doi:10.1002/9781118032497.ch10 fatcat:374tktuvgvekni4fnz3dgbytjm

DP-4-coloring of planar graphs with some restrictions on cycles [article]

Rui Li, Tao Wang
2019 arXiv   pre-print
It was originally used to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable.  ...  In this paper, we give three sufficient conditions for a planar graph is DP-4-colorable.  ...  This work was supported by the Natural Science Foundation of China and partially supported by the Fundamental Research Funds for Universities in Henan (YQPY20140051).  ... 
arXiv:1909.08511v1 fatcat:752mrabtpfcorpaitrvtrpxlzi
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