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Smaller planar triangle-free graphs that are not 3-list-colorable

A.N. Glebov, A.V. Kostochka, V.A. Tashkinov
2005 Discrete Mathematics  
In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices.  ...  The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors  ...  Later, Gutner [4] found a graph with the same properties having 164 vertices. In this note, we construct yet smaller examples of planar triangle-free graphs that are not 3-list-colorable.  ... 
doi:10.1016/j.disc.2004.10.015 fatcat:5grqreguyfcvbiogd2omcust2q

Edge Bounds and Degeneracy of Triangle-Free Penny Graphs and Squaregraphs

David Eppstein
2018 Journal of Graph Algorithms and Applications  
We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter $D=\Omega(\sqrt n)$, and at most $\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr  ...  However, not every triangle-free planar graph is 3-list-colorable: if each vertex is given a list of three colors, it is not always possible to assign each vertex a color from its list that differs from  ...  Grötzsch proved that these graphs are 3-colorable [21, 22] and they can be 3-colored in linear time [23] .  ... 
doi:10.7155/jgaa.00463 fatcat:2kr3rhq2cjcuflilbz5owptjzu

Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count [chapter]

David Eppstein
2018 Lecture Notes in Computer Science  
We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter D = Ω( √ n), and at most min 2n − Ω( √ n), 2n − D − 2 edges.  ...  However, not every triangle-free planar graph is 3-list-colorable: if each vertex is given a list of three colors, it is not always possible to assign each vertex a color from its list that differs from  ...  Grötzsch proved that these graphs are 3-colorable [21, 22] and they can be 3-colored in linear time [23] .  ... 
doi:10.1007/978-3-319-73915-1_39 fatcat:2scpbb7ckbajlogaxd53ox5zhq

Three-coloring triangle-free planar graphs in linear time (extended abstract) [chapter]

Zdeněk Dvořák, Ken-ichi Kawarabayashi, Robin Thomas
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
Grötzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors.  ...  We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement.  ...  There is a linear-time algorithm to 3color an input triangle-free planar graph.  ... 
doi:10.1137/1.9781611973068.127 fatcat:7j4cjqynhna5nayeq3uzwmigyu

List Multicoloring of Planar Graphs and Related Classes [article]

Glenn G. Chappell
2022 arXiv   pre-print
For general planar graphs, we show that if a/b < 42/5, then there exists a planar graph that is not (a:b)-choosable, thus improving on a result of X. Zhu, which had 42/9.  ...  We show that for positive integers a and b, every bipartite planar graph is (a:b)-choosable iff a/b≥ 3.  ...  We may assume that H is K 3 ; otherwise choose a vertex that lies in a triangle containing H, color this vertex using colors in its list that are not used on either vertex of H, and add the new vertex  ... 
arXiv:2205.09856v1 fatcat:wmf3tdec7jchvn3phftkeiafwa

Three-coloring triangle-free planar graphs in linear time [article]

Zdenek Dvorak, Ken-ichi Kawarabayashi, Robin Thomas
2013 arXiv   pre-print
Grotzsch's theorem states that every triangle-free planar graph is 3-colorable. Several relatively simple proofs of this fact were provided by Thomassen and other authors.  ...  We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement.  ...  Acknowledgement We are indebted to a referee for carefully reading the manuscript and for pointing out a couple of errors.  ... 
arXiv:1302.5121v1 fatcat:tm6tjk46bfgv5fjhiv7ph22egm

Exponentially Many 4-List-Colorings of Triangle-Free Graphs on Surfaces [article]

Tom Kelly, Luke Postle
2016 arXiv   pre-print
Thomassen proved that every planar graph G on n vertices has at least 2^n/9 distinct L-colorings if L is a 5-list-assignment for G and at least 2^n/10000 distinct L-colorings if L is a 3-list-assignment  ...  We prove the same result if G is triangle-free and L is a 4-list-assignment of G, where ϵ=1/8, and α= 130.  ...  Theorem 1.6 can not be extended to list-coloring, since there exist triangle-free planar graphs that are not 3-choosable.  ... 
arXiv:1602.04717v1 fatcat:2tra5lxlzjb6bckz4jqzfm6pgi

The Grötzsch Theorem for the Hypergraph of Maximal Cliques

Bojan Mohar, Riste Skrekovski
1999 Electronic Journal of Combinatorics  
We also extend this result to list colorings by proving that ${\cal H}(G)$ is 4-choosable for every planar or projective planar graph $G$.  ...  In this paper, we extend the Grötzsch Theorem by proving that the clique hypergraph ${\cal H}(G)$ of every planar graph is 3-colorable.  ...  We are grateful to anonymous referee who pointed out a "hidden" error in an earlier version of the paper.  ... 
doi:10.37236/1458 fatcat:mk3jt7rwfzdbpnkte4uya34na4

New Linear-Time Algorithms for Edge-Coloring Planar Graphs

Richard Cole, Łukasz Kowalik
2007 Algorithmica  
We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree Δ with max{Δ, 9} colors.  ...  Thus the coloring is optimal for graphs with maximum degree Δ ≥ 9. Moreover for Δ = 4, 5, 6 we give linear-time algorithms that use Δ + 2 colors.  ...  Vizing [8] showed that planar graphs with Δ ≥ 8 are in Class 1. He also noted that there are Class 2 planar graphs for Δ ∈ {2, 3, 4, 5}.  ... 
doi:10.1007/s00453-007-9044-3 fatcat:fi3i5d42jrgujfnakyd2ccweky

Algorithmic complexity of list colorings

Jan Kratochvíl, Zsolt Tuza
1994 Discrete Applied Mathematics  
in at most three sets L(u), (3) each vertex UE V has degree at most three, and (4) G is a planar graph.  ...  One of our results states that this decision problem remains NP-complete even if all of the following conditions are met: (1) each set L(u) has at most three elements, (2) each "color" x E UVEV L(u) occurs  ...  Proposition 3 . 3 The following variants of LC are polynomially solvable: Problem 4 . 4 Suppose that G = ( V, E) is a planar triangle-free graph with ) L(u)) = 3 for all v E V.  ... 
doi:10.1016/0166-218x(94)90150-3 fatcat:24a7u4vgufewhi34flnxcxbeei

Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8 [article]

Zdenek Dvorak, Luke Postle
2016 arXiv   pre-print
Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.  ...  We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring.  ...  For example, while every planar triangle-free graph is 3colorable, there exist such graphs that are not 3-choosable [17] , and while every planar graph is 4-colorable, not all are 4-choosable [16] .  ... 
arXiv:1508.03437v2 fatcat:iqjnicgb5vcthd2r3flfhifai4

Many 3-colorings of triangle-free planar graphs

Carsten Thomassen
2007 Journal of combinatorial theory. Series B (Print)  
That result implies that the number of 3-colorings of a planar triangle-free graph with n vertices is at least n/6.  ...  The 3-color matrix of G is the matrix whose rows are all these vectors. In [4] it was proved that the 3-color matrix of a planar graph has full column rank if and only if the graph is triangle-free.  ...  Acknowledgment Thanks are due to one of the referees who corrected a number of inaccuracies in the proof.  ... 
doi:10.1016/j.jctb.2006.06.005 fatcat:pdaoqmvmdzelfjfunf6uxsycwm

Three-coloring triangle-free planar graphs in linear time

Zdeněk Dvořák, Ken-Ichi Kawarabayashi, Robin Thomas
2011 ACM Transactions on Algorithms  
Grötzsch's theorem states that every triangle-free planar graph is 3-colorable, and several relatively simple proofs of this fact were provided by Thomassen and other authors.  ...  We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement.  ...  Theorem 1. 1 . 1 Every triangle-free planar graph is 3-colorable.  ... 
doi:10.1145/2000807.2000809 fatcat:awhjligusbahhdzuewcmelpxby

Note on 3-Choosability of Planar Graphs with Maximum Degree 4 [article]

François Dross and Borut Lužar and Mária Maceková and Roman Soták
2019 arXiv   pre-print
Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and  ...  Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete.  ...  Therefore it is not surprising that an equivalent of Grötzsch's result does not hold in this setting; as shown by Voigt [27] , there are triangle-free planar graphs which are not 3-choosable.  ... 
arXiv:1809.09347v2 fatcat:ptix3qmyxnhk7oobs7etwsaweq

Fast 3-coloring Triangle-Free Planar Graphs

Lukasz Kowalik
2009 Algorithmica  
We show the first o(n 2 ) algorithm for coloring vertices of triangle-free planar graphs using three colors. The time complexity of the algorithm is O(n log n).  ...  Our approach can be also used to design O(n polylog n)-time algorithms for two other similar coloring problems.  ...  Thanks go also to Maciej Kurowski for many interesting discussions inÅrhus, not only these on Grötzsch's theorem.  ... 
doi:10.1007/s00453-009-9295-2 fatcat:edrs6hmx2facbkofgdmtfnfd44
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