Formula omitted]-coloring of Kneser graphs

Abstract

A b -coloring of a graph G with k colors is a proper coloring of G using k 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 k for which G has a b -coloring using k colors is the b -chromatic number b ( G ) of G . The b -spectrum S b ( G ) of a graph G is the set of positive integers k , χ ( G ) ≤ k ≤ b ( G ) , for which G has a b -coloring using k colors. A graph G is b -continuous if S b ( G ) = the closed interval [ χ ( G ) , b ( G ) ] . In this paper, we obtain an upper bound for the b -chromatic number of some families of Kneser graphs. In addition we establish that [ χ ( G ) , n + k + 1 ] ⊂ S b ( G ) for the Kneser graph G = K ( 2 n + k , n ) whenever 3 ≤ n ≤ k + 1 . We also establish the b -continuity of some families of regular graphs which include the family of odd graphs.