Characterization of bγ-perfect graphs

Abstract

A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that is adjacent to a vertex of every other color, and the b-chromatic number b(G) of a graph G is the largest k such that G admits a b-coloring with k colors. A Grundy coloring is a proper coloring with integers 1, 2, … such that every vertex has a neighbor of each color smaller than its own color, and the Grundy number γ(G) of a graph G is the largest k such that G admits a Grundy coloring with k colors. An a-coloring (or complete coloring) is a proper coloring of the vertices of a graph such that the union of any two color classes is not an independent set, and the a-chromatic number ψ(G) of a graph G is the largest k such that G admits an a-coloring with k colors. A graph is bγ-perfect if b(H) = γ(H) holds for every induced subgraph of G. We study the relationship between b and γ and characterize bγ-perfect graphs as a special subclass of P4-free graphs. We also show how to compute b in polynomial time for every P4-free graph. We also characterize bψ-perfect graphs.