On the b-chromatic number of regular graphs

Abstract

The b-chromatic number of a graph G is the largest integer k , such that G admits a proper k -coloring in which every color class contains at least one vertex that has a neighbor in each of the other color classes. We prove that every d -regular graph with at least 2 d 3 vertices has b-chromatic number d + 1 , the b-chromatic number of an arbitrary d -regular graph with girth g = 5 is at least ⌊ d + 1 2 ⌋ and every d -regular graph, d ≥ 6 , with diameter at least d and with no 4-cycles, has b-chromatic number d + 1 .