Independent Sets in Tensor Products of Three Vertex-transitive Graphs

Abstract

The tensor product $T(G_{1},G_{2},G_{3})$ of graphs $G_{1}$, $G_{2}$ and $G_{3}$ is defined by \[ VT(G_{1},G_{2},G_{3}) = V(G_{1}) \times V(G_{2}) \times V(G_{3}) \] and \[ ET(G_{1},G_{2},G_{3}) = \{ [(u_{1},u_{2},u_{3}), (v_{1},v_{2},v_{3})]: |\{ i: (u_{i},v_{i}) \in E(G_{i}) \}| \geq 2 \}. \] 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 $T(G_{1}, G_2, G_3)$. So \[ \alpha T(G_{1},G_{2},G_{3}) \geq \max \{ \alpha(G_{1}) \alpha(G_{2}) |G_{3}|, \alpha(G_{1}) \alpha(G_{3}) |G_{2}|, \alpha(G_{2}) \alpha(G_{3}) |G_{1}| \}. \] In this paper, we prove that the equality holds if $G_{1}$ and $G_{2}$ are vertex-transitive graphs and $G_{3}$ is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of $T(G_{1}$, $G_{2}$, $G_{3})$ is determined.