Strong cliques in vertex-transitive graphs

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.