On the b-coloring of tight graphs

Abstract

A coloring c of a graph $G=(V,E)$ 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 $m-1$. A graph G is tight if it has exactly m(G) vertices of degree $m(G)-1$, and any other vertex has degree at most $m(G)-2$. In this paper, we show that the b-chromatic number of tight graphs with girth at least 8 is at least $m(G)-1$ and characterize the graphs G such that $b(G)=m(G)$. Lin and Chang (2013) conjectured that the b-chromatic number of any graph in $\mathcal {B}_{m}$ is m or $m-1$ where $\mathcal {B}_{m}$ is the class of tight bipartite graphs $(D,D{^\prime })$ of girth 6 such that D is the set of vertices of degree $m-1$. We verify the conjecture of Lin and Chang for some subclass of $\mathcal {B}_{m}$, and we give a lower bound for any graph in $\mathcal {B}_{m}$.