Investigating the [formula omitted]-chromatic number of bipartite graphs by using the bicomplement

Abstract

A b-coloring of a graph G by k colors is a proper vertex coloring such that every color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k−1 color classes. The b-chromatic number χb(G) is the maximum integer k for which G has a b-coloring by k colors. For a bipartite graph G=(A∪B,E), the bicomplement of G is the bipartite graph G˜=(A∪B,E˜) with E˜:={{a,b}∣a∈A,b∈B,{a,b}∉E}. In this paper, we investigate the b-chromatic number for bipartite graphs with a special bicomplement. In particular, we consider graphs G for which G˜ is disconnected or has maximum degree Δ(G˜)≤2. Moreover, we give partial answers to the question “Which d-regular bipartite graphs G satisfy χb(G)=d+1?” and we show a Nordhaus–Gaddum-type result for G and G˜.