The finite simple group k-valent GRR conjecture

A graphical regular representation (GRR) of a group $G$ is a Cayley graph whose full automorphism group is isomorphic to $G$; a GRR is $k$-valent when the corresponding graph has valency $k$. The finite simple group $k$-valent GRR conjecture. For each integer $k\geq3$, except for a finite number of cases, every finite nonabelian simple group has a $k$-valent GRR. This extends the previously known cubic case, but the source presents the general assertion as a conjecture and studies it for alternating groups and groups of Lie type.