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

Petr A. Golovach; Matthew Johnson; Daniël Paulusma; Jian Song

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arXiv · 1407.1482

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

Petr A. Golovach·Matthew Johnson·Daniël Paulusma·Jian Song

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Abstract

For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. 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.

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Petr A. Golovach, Matthew Johnson, Daniël Paulusma, Jian Song. 2016-02-15. A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs. https://arxiv.org/abs/1407.1482

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