The b -chromatic number of the cartesian product of two graphs

In this paper we study the b -chromatic number of the cartesian product of two graphs. The b -chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G , such that we obtain a proper coloring and each color i has at least one representative χi adjacent to a vertex of every color j , 1 ≦ j ≠ i ≦ k . In this paper we get ρ( G□H ) ≦ ρ( G )( nH + 1) + Δ( H ) + 1, when the girth of G is assumed to be greater than or equal to 7.