Fixed parameter algorithms for restricted coloring problems

TL;DR

This paper develops polynomial-time fixed parameter algorithms for various restricted coloring problems on specific graph classes, expanding tractability beyond NP-hard cases.

Contribution

Introduces fixed parameter algorithms for multiple coloring problems on $(q,q-4)$-graphs and related classes, a significant step beyond NP-hardness.

Findings

Algorithms are fixed parameter tractable on $q(G)$.

Connected $(q,q-4)$-graphs with at least $q$ vertices are 2-clique-colorable.

Acyclic colorings of cographs are also nonrepetitive.

Abstract

In this paper, we obtain polynomial time algorithms to determine the acyclic chromatic number, the star chromatic number, the Thue chromatic number, the harmonious chromatic number and the clique chromatic number of $P_4$-tidy graphs and $(q,q-4)$-graphs, for every fixed $q$. These classes include cographs, $P_4$-sparse and $P_4$-lite graphs. All these coloring problems are known to be NP-hard for general graphs. These algorithms are fixed parameter tractable on the parameter $q(G)$, which is the minimum $q$ such that $G$ is a $(q,q-4)$-graph. We also prove that every connected $(q,q-4)$-graph with at least $q$ vertices is 2-clique-colorable and that every acyclic coloring of a cograph is also nonrepetitive.