Independent Sets in Direct Products of Vertex-transitive Graphs

TL;DR

This paper proves a formula for the independence number of the direct product of vertex-transitive graphs and characterizes all maximum independent sets, resolving a longstanding open problem.

Contribution

It establishes the exact independence number for the direct product of vertex-transitive graphs and describes the structure of all maximum independent sets.

Findings

The independence number of G×H equals max{α(G)|H|, α(H)|G|} for all vertex-transitive graphs G and H.

Provides a complete characterization of maximum independent sets in G×H.

Confirms a conjecture posed by Tardif in 1998.

Abstract

The direct product $G\times H$ of graphs $G$ and $H$ is defined by: \[V(G\times H)=V(G)\times V(H)\] and \[E(G\times H)=\left\{[(u_1,v_1),(u_2,v_2)]: (u_1,u_2)\in E(G) \mbox{\ and\ } (v_1,v_2)\in E(H)\right\}.\] In this paper, we will prove that the equality $$\alpha(G\times H)=\max\{\alpha(G)|H|, \alpha(H)|G|\}$$ holds for all vertex-transitive graphs $G$ and $H$, which provides an affirmative answer to a problem posed by Tardif (Discrete Math. 185 (1998) 193-200). Furthermore, the structure of all maximum independent sets of $G\times H$ are determined.