Cubic Graphical Regular Representations of ${\mathrm{PSU}}_3(q)$

Abstract

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that ${\mathrm{PSU}}_3(q)$ has a cubic GRR if and only if $q\ge 4$. Moreover, a cubic GRR of ${\mathrm{PSU}}_3(q)$ is constructed for each of these $q$.