On the b-coloring of tight graphs

Abstract

A coloring c of a graph $$G=(V,E)$$G=(V,E) is a b-coloring if for every color i there is a vertex, say w(i), of color i whose neighborhood intersects every other color class. The vertex w(i) is called a b-dominating vertex of color i. The b-chromatic number of a graph G, denoted by b(G), is the largest integer k such that G admits a b-coloring with k colors. Let m(G) be the largest integer m such that G has at least m vertices of degree at least $$m-1$$m-1. A graph G is tight if it has exactly m(G) vertices of degree $$m(G)-1$$m(G)-1, and any other vertex has degree at most $$m(G)-2$$m(G)-2. In this paper, we show that the b-chromatic number of tight graphs with girth at least 8 is at least $$m(G)-1$$m(G)-1 and characterize the graphs G such that $$b(G)=m(G)$$b(G)=m(G). Lin and Chang (2013) conjectured that the b-chromatic number of any graph in $$\mathcal {B}_{m}$$Bm is m or $$m-1$$m-1 where $$\mathcal {B}_{m}$$Bm is the class of tight bipartite graphs $$(D,D{^\prime })$$(D,Dź) of girth 6 such that D is the set of vertices of degree $$m-1$$m-1. We verify the conjecture of Lin and Chang for some subclass of $$\mathcal {B}_{m}$$Bm, and we give a lower bound for any graph in $$\mathcal {B}_{m}$$Bm.