Vertex-transitive closures of graphs

A vertex-transitive closure of $\mathrm{\Gamma }$ is a vertex-transitive supergraph of $\mathrm{\Gamma }$ on the same vertex set. The vertex-transitive number of a graph $\mathrm{\Gamma }$, denoted by $d_{vt}(\mathrm{\Gamma })$, is the smallest integer for which there exists a $d_{vt}(\mathrm{\Gamma })$-regular vertex-transitive closure of $\mathrm{\Gamma }$. In this paper we use various algebraic and combinatorial methods to study vertex-transitive closures of graphs and the associated vertex-transitive number.