Maximization coloring problems on graphs with few [formula omitted

Abstract

Given a graph G, a greedy coloring of G is a proper coloring such that, for each two colors i<j, every vertex of G colored j has a neighbor colored i. The Grundy number is the maximum number of colors in a greedy coloring of G. Zaker (2006) proved that determining the Grundy number is NP-hard even for complements of bipartite graphs. A b-coloring of G is a proper coloring such that every color class contains a vertex which is adjacent to at least one vertex in every other color class. The b-chromatic number is the maximum number of colors in a b-coloring of G. Irving and Manlove (1999) proved that determining the b-chromatic number is NP-hard. In this paper, we obtain polynomial time algorithms to determine the Grundy number and the b-chromatic number of (q,q−4)-graphs, for every fixed q, which are the graphs such that every set of at most q vertices induces at most q−4 distinct P4. These algorithms are fixed parameter tractable on the parameter q(G), where q(G) is the minimum q such that G is a (q,q−4)-graph.