NP-completeness of list coloring and precoloring extension on the edges of planar graphs

A graph $G$ is $(d_1,...,d_l)$-colorable if the vertex set of $G$ can be partitioned into subsets $V_1,\ldots ,V_l$ such that the graph $G[V_i]$ induced by the vertices of $V_i$ has maximum degree at most $d_i$ for all $1 \leq i \leq l$. In this paper, we focus on complexity aspects of such colorings when $l=2,3$. More precisely, we prove that, for any fixed integers $k,j,g$ with $(k,j) \neq (0,0)$ and $g\geq3$, either every planar graph with girth at least $g$ is $(k,j)$-colorable or it is NP-complete to determine whether a planar graph with girth at least $g$ is $(k,j)$-colorable. Also, for any fixed integer $k$, it is NP-complete to determine whether a planar graph that is either $(0,0,0)$-colorable or non-$(k,k,1)$-colorable is $(0,0,0)$-colorable. Additionally, we exhibit non-$(3,1)$-colorable planar graphs with girth 5 and non-$(2,0)$-colorable planar graphs with girth 7.

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Adapted list coloring of planar graphs

Journal of Graph Theory ◽

10.1002/jgt.20391 ◽

2009 ◽

Vol 62 (2) ◽

pp. 127-138 ◽

Cited By ~ 8

Author(s):

Louis Esperet ◽

Mickaël Montassier ◽

Xuding Zhu

Keyword(s):

Planar Graphs ◽

List Coloring

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2-List-coloring planar graphs without monochromatic triangles

Journal of Combinatorial Theory Series B ◽

10.1016/j.jctb.2008.02.006 ◽

2008 ◽

Vol 98 (6) ◽

pp. 1337-1348 ◽

Cited By ~ 6

Author(s):

Carsten Thomassen

Keyword(s):

Planar Graphs ◽

List Coloring

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A method of graph reduction and its applications

Discrete Mathematics and Applications ◽

10.1515/dma-2018-0022 ◽

2018 ◽

Vol 28 (4) ◽

pp. 249-258

Author(s):

Dmitrii V. Sirotkin ◽

Dmitriy S. Malyshev

Keyword(s):

Planar Graphs ◽

Independent Set ◽

Simple Graph ◽

Vertex Degree ◽

Graph Reduction ◽

Independent Set Problem ◽

Np Completeness ◽

Maximal Vertex

Abstract The independent set problem for a given simple graph is to determine the size of a maximal set of its pairwise non-adjacent vertices. We propose a new way of graph reduction leading to a new proof of the NP-completeness of the independent set problem in the class of planar graphs and to the proof of NP-completeness of this problem in the class of planar graphs having only triangular internal facets of maximal vertex degree 18.

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Distributed Coloring in Sparse Graphs with Fewer Colors

The Electronic Journal of Combinatorics ◽

10.37236/8395 ◽

2019 ◽

Vol 26 (4) ◽

Author(s):

Pierre Aboulker ◽

Marthe Bonamy ◽

Nicolas Bousquet ◽

Louis Esperet

Keyword(s):

Planar Graph ◽

Efficient Algorithm ◽

Distributed Algorithm ◽

Planar Graphs ◽

List Coloring ◽

Sparse Graphs ◽

Four Color Theorem

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof gives a (sequential) quadratic algorithm finding such a coloring. A natural problem is to improve this complexity in the distributed setting. Using the fact that planar graphs contain linearly many vertices of degree at most 6, Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we show how to color planar graphs with 6 colors in $\text{polylog}(n)$ rounds. Our algorithm indeed works more generally in the list-coloring setting and for sparse graphs (for such graphs we improve by at least one the number of colors resulting from an efficient algorithm of Barenboim and Elkin, at the expense of a slightly worst complexity). Our bounds on the number of colors turn out to be quite sharp in general. Among other results, we show that no distributed algorithm can color every $n$-vertex planar graph with 4 colors in $o(n)$ rounds.

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Coloring Problems on Bipartite Graphs of Small Diameter

The Electronic Journal of Combinatorics ◽

10.37236/9931 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Victor A. Campos ◽

Guilherme C.M. Gomes ◽

Allen Ibiapina ◽

Raul Lopes ◽

Ignasi Sau ◽

...

Keyword(s):

Small Diameter ◽

Bipartite Graphs ◽

List Coloring ◽

Coloring Problem ◽

Np Completeness ◽

Np Complete

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].

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