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The complexity of planar graph choosability

Shai Gutner
1996 Discrete Mathematics  
We consider the complexity of deciding whether a given graph is k-choosable for some constant k.  ...  In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable.  ...  In the present paper we consider the complexity of deciding whether a given graph is k-choosable for some constant k.  ... 
doi:10.1016/0012-365x(95)00104-5 fatcat:4btk53rlufaunc5x7hsugdonoi

The complexity of planar graph choosability [article]

Shai Gutner
2008 arXiv   pre-print
We consider the complexity of deciding whether a given graph is k-choosable for some constant k.  ...  In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable.  ...  In the present paper we consider the complexity of deciding whether a given graph is k-choosable for some constant k.  ... 
arXiv:0802.2668v1 fatcat:4lkaviaj6rh3zbins3vj32wlbi

Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs [article]

Konrad K. Dabrowski, Francois Dross, Matthew Johnson, Daniel Paulusma
2019 arXiv   pre-print
By using known examples of non-3-choosable and non-4-choosable graphs, this enables us to classify the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar  ...  We also classify the complexity of k-Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.  ...  We thank Zdeněk Dvořák and Steven Kelk for helpful comments on earlier versions of this paper.  ... 
arXiv:1506.06564v5 fatcat:gpljlpxywfb3zd4hk7ebwpeihy

Filling the complexity gaps for colouring planar and bounded degree graphs

Konrad K. Dabrowski, François Dross, Matthew Johnson, Daniël Paulusma
2019 Journal of Graph Theory  
We give a complete classification of the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and  ...  We also give a complete classification of the complexity of this problem and a number of related colouring problems for graphs with bounded maximum degree.  ...  We thank Steven Kelk for helpful comments on an earlier version of this paper.  ... 
doi:10.1002/jgt.22459 fatcat:lugoxznu5ngbnebcyrxsaghpbe

Page 3442 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
We consider the complexity of deciding whether a given graph is k-choosable for some constant k.  ...  In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable.  ... 

Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs [chapter]

Konrad K. Dabrowski, François Dross, Matthew Johnson, Daniël Paulusma
2016 Lecture Notes in Computer Science  
We give a complete classification of the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and  ...  We also give a complete classification of the complexity of this problem and a number of related colouring problems for graphs with bounded maximum degree.  ...  We thank Steven Kelk for helpful comments on an earlier version of this paper.  ... 
doi:10.1007/978-3-319-29516-9_9 fatcat:aufsai6bmzcefe4qezbd3wbjfa

A refinement of choosability of graphs [article]

Xuding Zhu
2019 arXiv   pre-print
For the other partitions of k sandwiched between {k} and {1,1,..., 1} in terms of refinements, λ-choosability reveals a complex hierarchy of colourability of graphs.  ...  In particular, it is proved that a conjecture of Kündgen and Ramamurthi on list colouring of planar graphs is implied by the conjecture that every planar graph is {2,2}-choosable, and also implied by the  ...  of non-4-choosable planar graphs.  ... 
arXiv:1811.08587v2 fatcat:24rnovkq7ndbjjd2n6lpkzobee

Complexity of choosability with a small palette of colors [article]

Marc Demange, Dominique de Werra
2017 arXiv   pre-print
We study complexity issues of choosability of graphs when the number k of colors is limited.  ...  We also exhibit some classes of graphs (defined by structural properties of their blocks) which are choosable.  ...  Since {2, 3}-CH is Π p 2 -complete in bipartite graphs Some remarks about complexity of choosability We conclude this work with a few remarks on the complexity of choosability.  ... 
arXiv:1601.01768v2 fatcat:xk57pcfvc5edho7good4b6uytm

Choosability in bounded sequential list coloring [article]

Simone Gama, Rosiane de Freitas, Mário Salvatierra
2018 arXiv   pre-print
some classes of graphs, such as complete bipartite graph, which is 3 -choosable, but 2 -(γ,μ)-choosable.  ...  This is a Π_2^P-complete problem, however, we show that k-(γ,μ)-choosability is an NP-problem due to its relation with the k-coloring of a graph and application of methods of proof in choosability for  ...  This method is used to show that every planar graph is 6-choosable and every triangle-free planar graph are 4-choosable [17] .  ... 
arXiv:1812.11685v1 fatcat:c2bryajixjdjrhhgfxsfugtg4i

Graph Planarity Testing with Hierarchical Embedding Constraints [article]

Giuseppe Liotta, Ignaz Rutter, Alessandra Tappini
2019 arXiv   pre-print
In particular, we apply our techniques to the study of NodeTrix planarity testing of clustered graphs.  ...  We show that NodeTrix planarity testing with fixed sides is fixed-parameter tractable when parameterized by the size of the clusters and by the treewidth of the multi-graph obtained by collapsing the clusters  ...  Complexity of FPQ-choosable Planarity Testing As we are going to show, FPQ-Choosable Planarity Testing is fixedparameter tractable when parameterized by treewidth and number of FPQ-trees per vertex.  ... 
arXiv:1904.12596v3 fatcat:takkfqjfj5bbharavej4qr62wa

Choosability on H-free graphs

Petr A. Golovach, Pinar Heggernes, Pim van 't Hof, Daniël Paulusma
2013 Information Processing Letters  
We determine the computational complexity of the Choosability problem restricted to H-free graphs for all but four graphs H.  ...  any format or medium without the formal permission of the copyright holders.  ...  This paper is supported by EPSRC (EP/G043434/1) , Royal Society (JP100692), the Research Council of Norway (197548/F20), and by the European Research Council (267959).  ... 
doi:10.1016/j.ipl.2012.12.003 fatcat:7yp5k6karzditcgqtyrjwtsola

On an early paper of Maryam Mirzakhani [article]

William J. Martin
2017 arXiv   pre-print
This short note remembers her 1996 article in the Bulletin of the Institute of Combinatorics and its Applications and her early years as a mathematician.  ...  Maryam Mirzakhani, the first female (and first Iranian) Fields Medalist, passed away on July 14, 2017 at the age of 40.  ...  Saieed Akbari, Ebad Mahmoodian, Jeff Dinitz and the referee all provided valuable comments and corrections on early drafts of the manuscript.  ... 
arXiv:1709.07540v2 fatcat:e3mxmw6tyjcszkofr6zbly6bha

A note on choosability with defect 1 of graphs on surfaces [article]

Vida Dujmović, Djedjiga Outioua
2018 arXiv   pre-print
For example, the chromatic number of the family of toroidal graphs is known to be 7. The bound above implies that toroidal graphs are 5-choosable with defect 1.  ...  This note proves that every graph of Euler genus μ is 2 + √(3μ + 3) --choosable with defect 1 (that is, clustering 2).  ...  For example, Voigt proved [13] that there are planar graphs that are not 4-choosable, but Cushing and Kierstead [7] proved that planar graphs are 4-choosable if defect 1 is allowed, thereby answering  ... 
arXiv:1806.06149v1 fatcat:edcaabggpzcsjkqwojnyinemd4

Graph colorings with local constraints -- a survey

Zsolt Tuza
1997 Discussiones Mathematicae Graph Theory  
In one of the most general formulations, a graph G = (V, E), sets L(v) of admissible colors, and natural numbers c v for the vertices v ∈ V are given, and the question is whether there can be chosen a  ...  The particular case of constant |L(v)| with c v = 1 for all v ∈ V leads to the concept of choice number, a graph parameter showing unexpectedly different behavior compared to the chromatic number, despite  ...  (vii) Every planar graph of girth 5 is 3-choosable (Thomassen [177] ). (viii) Every bipartite planar graph is 3-choosable (Alon, Tarsi [7] ).  ... 
doi:10.7151/dmgt.1049 fatcat:du2wuuygubedxp5jzztrmw5ike

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Petr A. Golovach, Matthew Johnson, Daniël Paulusma, Jian Song
2016 Journal of Graph Theory  
2016) 'A survey on the computational complexity of coloring graphs with forbidden subgraphs.', Journal of graph theory. .  ...  Further information on publisher's website: http://dx.(2016) 'A survey on the computational complexity of coloring graphs with forbidden subgraphs.  ...  The research leading to these results has received funding from EPSRC (EP/G043434/1) and the European Research Acknowledgments.  ... 
doi:10.1002/jgt.22028 fatcat:mv5j3zmgjjfkhij5inlnbxdxhy
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