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

2007
*
Discrete Mathematics
*

Steinberg's question from 1975 whether every

doi:10.1016/j.disc.2005.11.041
fatcat:r6xnyme46bh4fik7smml6uvrzy
*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 . ...##
###
The proof of Steinberg's three coloring conjecture
[article]

2006
*
arXiv
*
pre-print

The well-known Steinberg's conjecture asserts that any

arXiv:math/0607497v1
fatcat:5nplm5exifcbjdzudt6lthbnsa
*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. ...##
###
Colorings of plane graphs without long monochromatic facial paths

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}. ...

##
###
Steinberg's Conjecture is false

2017
*
Journal of combinatorial theory. Series B (Print)
*

Steinberg conjectured in 1976 that every

doi:10.1016/j.jctb.2016.07.006
fatcat:h2unnoaqxraxlfcn2ipkc2f5ri
*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. ...##
###
Steinberg's Conjecture is false
[article]

2016
*
arXiv
*
pre-print

Steinberg conjectured in 1976 that every

arXiv:1604.05108v2
fatcat:23r5or7ijjc2jmlvrmzk7caffm
*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. ...##
###
Locally planar toroidal graphs are $5$-colorable

1982
*
Proceedings of the American Mathematical Society
*

Essentially this hypothesis means that small neighborhoods

doi:10.1090/s0002-9939-1982-0640251-3
fatcat:g6m5mk4xurbwpmdk7lpokvda3a
*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. ...##
###
Locally Planar Toroidal Graphs are 5-Colorable

1982
*
Proceedings of the American Mathematical Society
*

Essentially this hypothesis means that small neighborhoods

doi:10.2307/2043580
fatcat:udhyykyzwzhrxf6f2vt5h4uuku
*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. ...##
###
Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph
[article]

2012
*
arXiv
*
pre-print

The famous

arXiv:0911.1587v3
fatcat:ifjdqgpvjnh7nlitqzr26p2hcq
*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. ...##
###
Planar graphs without cycles of length 4, 5, 8, or 9 are 3-choosable

2010
*
Discrete Mathematics
*

It is shown that a

doi:10.1016/j.disc.2009.08.005
fatcat:h32fm763gjfohjgkbysnibbena
*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*? ...##
###
Some counterexamples associated with the three-color problem

1980
*
Journal of combinatorial theory. Series B (Print)
*

In this paper we construct some counterexamples

doi:10.1016/0095-8956(80)90051-9
fatcat:xoxmwtbuknfathgmnunckzgvi4
*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. ...##
###
Planar graphs without adjacent cycles of length at most seven are 3-colorable

2010
*
Discrete Mathematics
*

We prove that every

doi:10.1016/j.disc.2009.08.010
fatcat:fo2zunw57fduxgwrtsxpdfyfge
*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*. ...##
###
The Four Color Theorem – A New Simple Proof by Induction
[article]

2021
*
arXiv
*
pre-print

We consider, in our proof, possible

arXiv:1701.03511v5
fatcat:lox3xrgitfcsfg52e6dxptxwcm
*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. ...##
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Planar graphs without 4-,5- and 8-cycles are 3-colorable

2011
*
Discussiones Mathematicae Graph Theory
*

In this paper we prove that every

doi:10.7151/dmgt.1579
fatcat:mkoczhcgjzdk5mjnyt42s2kl3i
*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*. ...##
###
Algorithms
[chapter]

2011
*
Graph Coloring Problems
*

Density

doi:10.1002/9781118032497.ch10
fatcat:374tktuvgvekni4fnz3dgbytjm
*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 ...##
###
DP-4-coloring of planar graphs with some restrictions on cycles
[article]

2019
*
arXiv
*
pre-print

It was originally used to solve a longstanding conjecture by Borodin, stating that every

arXiv:1909.08511v1
fatcat:752mrabtpfcorpaitrvtrpxlzi
*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). ...
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