Primitivity and Independent Sets in Direct Products of Vertex-Transitive Graphs

TL;DR

This paper explores the primitivity of independent sets in vertex-transitive graphs, establishing new links between primitivity and maximum independent set structures in their direct products, and solving an open problem in graph theory.

Contribution

It introduces the concept of primitivity for independent sets and connects it to the structure of maximum independent sets in direct products of vertex-transitive graphs, solving an open problem.

Findings

Established the relationship between primitivity and maximum independent sets.

Solved an open problem regarding independent sets in powers of vertex-transitive graphs.

Provided new insights into the structure of independent sets in graph products.

Abstract

We introduce the concept of the primitivity of independent set in vertex-transitive graphs, and investigate the relationship between the primitivity and the structure of maximum independent sets in direct products of vertex-transitive graphs. As a consequence of our main results, we positively solve an open problem related to the structure of independent sets in powers of vertex-transitive graphs.