The 6-coloring and 6-continuity of the cartesian product of some graphs

Abstract

The 6-chromatic number ℓ(G) of a graph G is the maximum number k of colors that can be used to give a proper coloring of G, such that in each color class there exists a vertex having neighbors in all other k - 1 color classes. A graph G is 6-continuous if for every k, x(G) ≤ k ≤ ℓ(G), there exists a 6-coloring of this graph by k colors. In this paper, we study the 6-coloring and the 6-continuity of the graphs K_n □ K_1,P, K_n □ K_{2, p} and K_n □ K_P,P.