Shift-transitive and uniquely shift-transitive graphs of small degrees

Abstract

An automorphism $\sigma$ of a finite simple graph $\Gamma$ is a shift,
if for every vertex $v\in V(\Gamma), \sigma v$ is adjacent to $v$ in $\Gamma$.
The graph $\Gamma$ is shift-transitive, if for every pair of vertices $u, v\in V(\Gamma)$ there exists a sequence of shifts $\sigma_1, \sigma_2, ..., \sigma_k\in \Aut(\Gamma)$ such that $\sigma_{1}\sigma_{2}...\sigma_{k}u=v$.
If, in addition, for every pair of adjacent vertices $u, v\in V( \Gamma)$ there exists exactly one shift $\sigma\in \Aut(\Gamma)$ sending $u$ to $v$, then $\Gamma$ is uniquely shift-transitive.

The purpose of this paper is to prove that, if $\Gamma$ is a uniquely shift-transitive graph of valency at most 5 and $S_{\Gamma}$ is the set of shifts in $\Gamma$ then $<S_{\Gamma}>$ is an abeliansubgroup of $\Aut(\Gamma).$