On bb-coloring of the Kneser graphs

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