Restricted coloring problems on Graphs with few P 4’s

Abstract

In this paper, we obtain linear time algorithms to determine the acyclic chromatic number, the star chromatic number, the non repetitive chromatic number and the clique chromatic number of P 4-tidy graphs and (q, q − 4)-graphs, for every fixed q, which are the graphs such that every set with at most q vertices induces at most q − 4 distinct P 4’s. These classes include cographs and P 4-sparse graphs. We also obtain a linear time algorithm to compute the harmonious chromatic number of connected P 4-tidy graphs and connected (q, q − 4)-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, generalizing the main result of Lyons (2011).