b-chromatic number of cartesian product of Km with some particular graphs

The [Formula: see text]-chromatic number of a graph [Formula: see text], [Formula: see text], is the maximum number [Formula: see text] of colors that can be used to color the vertices of [Formula: see text], such that we obtain a proper coloring and each color [Formula: see text], [Formula: see text] has at least one representant vertex [Formula: see text] adjacent to a vertex of every color [Formula: see text], [Formula: see text]. We denote by [Formula: see text] the complete bipartite graph [Formula: see text] and [Formula: see text] the graph wheel, constituted by an [Formula: see text]-cycle with another vertex adjacent to all the vertices of the cycle. In this paper, we first determine the value of [Formula: see text], then [Formula: see text] under some conditions on the two integers [Formula: see text] and [Formula: see text].