Pacific Institute for the Mathematical Sciences - PIMS

Automorphisms of Cayley graphs that respect partitions

A Cayley graph $\Gamma={\rm Cay}(G;S)$ on a group $G$ with connection set $S$, is a graph whose vertices are labelled with the elements of $G$, with vertices $g_1$ and $g_2$ adjacent if $g_1^{-1}g_2 \in S$.  We say that an automorphism $\alpha$ of $\Gamma$ respects the partition $\mathcal C$ of the edge set of $\Gamma$ if for every $C \in \mathcal C$, we have $\alpha(C) \in \mathcal C$. I will discuss some obvious partitions of the edge set of a Cayley graph $\Gamma$, and find conditions under which a graph automorphism of $\Gamma$ that respects these partitions and fixes a vertex, must be an automorphism of the group $G$.