bb-coloring of Kneser graphs

A bb-coloring of a graph GG with kk colors is a proper coloring of GG using kk colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer kk for which GG has a bb-coloring using kk colors is the bb-chromatic number b(G)b(G) of GG. The bb-spectrum Sb(G)Sb(G) of a graph GG is the set of positive integers k,χ(G)≤k≤b(G)k,χ(G)≤k≤b(G), for which GG has a bb-coloring using kk colors. A graph GG is bb-continuous if Sb(G)Sb(G) = the closed interval [χ(G),b(G)][χ(G),b(G)]. In this paper, we obtain an upper bound for the bb-chromatic number of some families of Kneser graphs. In addition we establish that [χ(G),n+k+1]⊂Sb(G)[χ(G),n+k+1]⊂Sb(G) for the Kneser graph G=K(2n+k,n)G=K(2n+k,n) whenever 3≤n≤k+13≤n≤k+1. We also establish the bb-continuity of some families of regular graphs which include the family of odd graphs.