On the number of generators of groups acting arc-transitively on graphs

Abstract

Given a finite connected graph $\Gamma$ and a group $G$ acting transitively on the vertices of $\Gamma$, we prove that the number of vertices of $\Gamma$ and the cardinality of $G$ are bounded above by a function depending only on the cardinality of $\Gamma$ and on the exponent of $G$. We also prove that the number of generators of a group $G$ acting transitively on the arcs of a finite graph $\Gamma$ cannot be bounded by a function of the valency alone.