On -coloring of the Kneser graphs

A b -coloring of a graph G by k colors is a proper k -coloring of G such that in each color class there exists a vertex having neighbors in all the other k − 1 color classes. The b -chromatic number of a graph G , denoted by φ ( G ) , is the maximum k for which G has a b -coloring by k colors. It is obvious that χ ( G ) ≤ φ ( G ) . A graph G is b -continuous if for every k between χ ( G ) and φ ( G ) there is a b -coloring of G by k colors. In this paper, we study the b -coloring of Kneser graphs K ( n , k ) and determine φ ( K ( n , k ) ) for some values of n and k . Moreover, we prove that K ( n , 2 ) is b -continuous for n ≥ 17 .