Structure of independent sets in direct products of some vertex-transitive graphs

Abstract

Let Circ(r, n) be a circular graph. It is well known that its independence number α(Circ(r, n)) = r. In this paper we prove that $$\alpha (Circ(r,n) \times H) = \max \{ r|H|,n\alpha (H)\} $$ for every vertex transitive graph H, and describe the structure of maximum independent sets in Circ(r, n) × H. As consequences, we prove $$\alpha (G \times H) = \max \{ \alpha (G)|V(H)|,\alpha (H)|V(G)|\} $$ for G being Kneser graphs, and the graphs defined by permutations and partial permutations, respectively. The structure of maximum independent sets in these direct products is also described.