B-Coloring Graphs with Girth at Least 8

Abstract

A b-coloring of a graph is a proper coloring of its vertices such that every color class contains a vertex that has neighbors in all other color classes. The b-chromatic number of a graph is the largest integer b(G) such that the graph has a b-coloring with b(G) colors. This metric is upper bounded by the largest integer m (G) for which G has at least m (G) vertices with degree at least m (G) —1. There are a number of results reporting that graphs with high girth have high b-chromatic number when compared to m(G). Here, we prove that every graph with girth at least 8 has b-chromatic number at least m(G) 3 - 1. This proof is constructive and yields a polynomial time algorithm to find the b-chromatic number of G. Furthermore, we improve known partial results related to reducing the girth requirement of our proof.