Notes on the independence number in the cartesian product of graphs

Abstract

Every connected graph G with radius r(G) and independence number α(G) obeys α(G) ≥ r(G). Recently the graphs for which equality holds have been classified. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H, α(GH) = r(GH) if and only if one factor is a complete graph on two vertices, and the other is a nontrivial complete graph. We also prove a new (polynomial computable) lower bound α(GH) ≥ 2r(G)r(H) for the independence number and we classify graphs for which equality holds.