Bounds for the b-chromatic Number of Induced Subgraphs and G − e

Abstract

All graphs considered in this paper are simple, finite and undirected. Let G be a graph with vertex set V and edge set E. The order of G will be denoted by n and size (number of edges) by m. A b-coloring of a graph is a proper coloring of the vertices of G such that each color class contains a color dominating vertex (c.d.v.), that is, a vertex adjacent to at least one vertex of every other color class. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. A b-chromatic coloring of G denotes a b-coloring using b(G) colors. From the definition of χ(G), we observe that each color class of a χ-coloring contains a c.d.v. Thus ω(G) ≤ χ(G) ≤ b(G), where ω(G) is the size of a maximum clique of G.