Strong cliques in vertex-transitive graphs

TL;DR

This paper characterizes strong cliques in vertex-transitive graphs, classifies low-valency cases, and explores the relationship between strong cliques, CIS, and localizability, including counterexamples to previous assumptions.

Contribution

It provides a characterization of strong cliques in vertex-transitive graphs and classifies graphs with valency up to 4, advancing understanding of their structural properties.

Findings

Strong cliques satisfy |C||I|=|V| for all maximal independent sets I.

Vertex-transitive CIS graphs are characterized by the existence of strong cliques and independent sets.

An example of a vertex-transitive CIS graph that is not localizable is provided.

Abstract

A clique (resp., independent set) in a graph is strong if it intersects every maximal independent sets (resp., every maximal cliques). A graph is CIS if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique $C$ in a vertex-transitive graph $\Gamma$ is strong if and only if $|C||I|=|V(\Gamma)|$ for every maximal independent set $I$ of $\Gamma$. Based on this result we prove that a vertex-transitive graph is CIS if and only if it admits a strong clique and a strong independent set. We classify all vertex-transitive graphs of valency at most 4 admitting a strong clique, and give a partial characterization of $5$-valent vertex-transitive graphs admitting a strong clique. Our results imply that every vertex-transitive graph of valency at most $5$ that admits a strong clique is localizable. We answer an open question by providing an example of a vertex-transitive CIS graph which is not localizable.