Independence and coloring properties of direct products of some vertex-transitive graphs

Let α(G)α(G) and χ(G)χ(G) denote the independence number and chromatic number of a graph G  , respectively. Let G×HG×H be the direct product graph of graphs G and H. We show that if G and H   are circular graphs, Kneser graphs, or powers of cycles, then α(G×H)=max{α(G)|V(H)|,α(H)|V(G)|}α(G×H)=max{α(G)|V(H)|,α(H)|V(G)|} and χ(G×H)=min{χ(G),χ(H)}χ(G×H)=min{χ(G),χ(H)}.