Open Problems on Graph Coloring for Special Graph Classes

Paulusma, Daniël

doi:10.1007/978-3-662-53174-7_2

Cited by 26 publications

(22 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We remark that an extended version of this article, which includes a survey on computational complexity results for graph classes defined by some other forbidden pattern, is available . For a survey on computational complexity results and open problems for coloring graphs characterized by more than two forbidden induced subgraphs or for which some graph parameter is bounded, see .…”

Section: Introductionmentioning

confidence: 99%

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

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

Self Cite

130

160

View full text Add to dashboard Cite

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. For a positive integer k, a k-colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . . , k} such that c(u) = c(v) whenever uv ∈ E. The COLOURING problem is to decide, for a given G and k, whether a k-colouring of G exists. If k is fixed (that is, it is not part of the input), we have the decision problem k-COLOURING instead. We survey known results on the computational complexity of COLOURING and k-COLOURING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex. Finally, we also survey results for graph classes defined by some other forbidden pattern.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Golovach

Johnson

Paulusma

et al. 2016

Journal of Graph Theory

Self Cite

130

160

View full text Add to dashboard Cite

show abstract

“…The coloring problems investigated in this article are formally defined below, adopting the notation from other papers; see e.g. [32]. In each of the problems defined below, we consider k to be a fixed integer with k 1.…”

Section: Definitions and Notationmentioning

confidence: 99%

“…A long-standing and still open question about coloring problems is whether one can decide in polynomial time if a graph with diameter two can be properly colored using at most three colors (see e.g. [22,32]), and only recently the answer for the 3-Coloring problem on graphs with diameter at most three has been settled negatively by Mertzios and Spirakis [30]. Here we propose the investigation of analogous questions concerning list colorings of bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

Coloring Problems on Bipartite Graphs of Small Diameter

Campos

Gomes

Ibiapina

et al. 2021

Electron. J. Combin.

View full text Add to dashboard Cite

We investigate a number of coloring problems restricted to bipartite graphs with bounded diameter. First, we investigate the $k$-List Coloring, $k$-Coloring, and $k$-Precoloring Extension problems on bipartite graphs with diameter at most $d$, proving $\textsf{NP}$-completeness in most cases, and leaving open only the List $3$-Coloring and $3$-Precoloring Extension problems when $d=3$. Some of these results are obtained $\textsc{through}$ a proof that the Surjective $C_6$-Homomorphism problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four. Although the latter result has been already proved [Vikas, 2017], we present ours as an alternative simpler one. As a byproduct, we also get that $3$-Biclique Partition is $\textsf{NP}$-complete. An attempt to prove this result was presented in [Fleischner, Mujuni, Paulusma, and Szeider, 2009], but there was a flaw in their proof, which we identify and discuss here. Finally, we prove that the $3$-Fall Coloring problem is $\textsf{NP}$-complete on bipartite graphs with diameter at most four, and prove that $\textsf{NP}$-completeness for diameter three would also imply $\textsf{NP}$-completeness of $3$-Precoloring Extension on diameter three, thus closing the previously mentioned open cases. This would also answer a question posed in [Kratochvíl, Tuza, and Voigt, 2002].

show abstract

“…The first two problems we study are the following ones stated by Golovach et al [19] (see also [24]) who asked to determine their parameterized complexity. These questions were motivated by a result of Cai [8] who showed that Coloring Clique Modulator (the special case of Pre-Coloring Extension Clique Modulator when X = ∅) is fixed-parameter tractable (FPT).…”

Section: Introductionmentioning

confidence: 99%

Parameterized Pre-Coloring Extension and List Coloring Problems

Gutin¹,

Majumdar²,

Ordyniak³

et al. 2021

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Golovach, Paulusma and Song (Inf. Comput. 2014) asked to determine the parameterized complexity of the following problems parameterized by k: (1) Given a graph G, a clique modulator D (a clique modulator is a set of vertices, whose removal results in a clique) of size k for G, and a list L(v) of colors for every v ∈ V (G), decide whether G has a proper list coloring; (2) Given a graph G, a clique modulator D of size k for G, and a pre-coloring λP : X → Q for X ⊆ V (G), decide whether λP can be extended to a proper coloring of G using only colors from Q. For Problem 1 we design an O * (2 k )-time randomized algorithm and for Problem 2 we obtain a kernel with at most 3k vertices. Banik et al. (IWOCA 2019) proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph G, an integer k, and a list L(v) of exactly n − k colors for every v ∈ V (G), decide whether there is a proper list coloring for G. We obtain a kernel with O(k 2 ) vertices and colors and a compression to a variation of the problem with O(k) vertices and O(k 2 ) colors.

show abstract

Open Problems on Graph Coloring for Special Graph Classes

Cited by 26 publications

References 66 publications

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

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

Coloring Problems on Bipartite Graphs of Small Diameter

Parameterized Pre-Coloring Extension and List Coloring Problems

Contact Info

Product

Resources

About