On the b-chromatic number of cartesian products

We consider the b-chromatic number of cartesian products of graphs. We show that the b-chromatic number of Kn□d for d≥3 is one more than the degree; for d≥12 this follows from a result of Kratochvíl, Tuza and Voigt. We show that Km□Kn has b-chromatic number at most its degree, and give different approaches that come close to this bound. We also consider cartesian powers of general graphs, and show that the cartesian product of d graphs each with b-chromatic number n is at least d(n−1)+1. This extends a theorem of Kouider and Mahéo by removing their condition on independent sets as long as the factor graphs all have the same b-chromatic number.