On the b-coloring of P4-tidy graphs

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by @g"b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every t=@g(G),...,@g"b(G), and it is b-monotonic if @g"b(H"1)>=@g"b(H"2) for every induced subgraph H"1 of G, and every induced subgraph H"2 of H"1. In this work, we prove that P"4-tidy graphs (a generalization of many classes of graphs with few induced P"4s) are b-continuous and b-monotonic. Furthermore, we describe a polynomial time algorithm to compute theb-chromatic number for this class of graphs.