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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.  ...

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

David Eppstein
2018 Journal of Graph Algorithms and Applications

### 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}.  ...

### 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.  ...

### 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] .  ...

### 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.  ...

### 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.  ...

### 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.  ...

### 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.  ...
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