Independent Sets in Tensor Products of Three Vertex-transitive Graphs

Abstract

The tensor product T (G1, G2, G3) of graphs G1, G2 and G3 is defined by V T (G1, G2, G3) = V (G1)× V (G2)× V (G3) and ET (G1, G2, G3) = {[(u1, u2, u3), (v1, v2, v3)] : |{i : (ui, vi) ∈ E(Gi)}| ≥ 2}. From this definition, it is easy to see that the preimage of the direct product of two independent sets of two factors under projections is an independent set of T (G1, G2, G3). So αT (G1, G2, G3) ≥ max{α(G1)α(G2)|G3|, α(G1)α(G3)|G2|, α(G2)α(G3)|G1|}. In this paper, we prove that the equality holds if G1 and G2 are vertex-transitive graphs and G3 is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of T (G1, G2, G3) is determined.