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A non-3-choosable planar graph without cycles of length 4 and 5
2007
Discrete Mathematics
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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]
2006
arXiv
pre-print
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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
2020
Discussiones Mathematicae Graph Theory
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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
2017
Journal of combinatorial theory. Series B (Print)
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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]
2016
arXiv
pre-print
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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
1982
Proceedings of the American Mathematical Society
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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
1982
Proceedings of the American Mathematical Society
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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]
2012
arXiv
pre-print
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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
2010
Discrete Mathematics
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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
1980
Journal of combinatorial theory. Series B (Print)
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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
2010
Discrete Mathematics
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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]
2021
arXiv
pre-print
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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
2011
Discussiones Mathematicae Graph Theory
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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
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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]
2019
arXiv
pre-print
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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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