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

TL;DR

This survey reviews the computational complexity of graph coloring problems, especially for classes defined by forbidden subgraphs, highlighting known results, variants, and open questions in the field.

Contribution

It compiles and analyzes existing complexity results for graph coloring with forbidden subgraphs, providing a comprehensive overview and identifying gaps in current knowledge.

Findings

Complexity classifications vary based on forbidden subgraph structures.

Certain graph classes allow polynomial-time coloring algorithms.

Open problems remain for specific forbidden subgraph configurations.

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.