10 problems

Babai–Godsil–Imrich–Lovász conjecture. The proportion of inverse-closed subsets $S$ of $R$ such that $\operatorname{Cay}(R,S)$ is a GRR of $R$ approaches $1$ as $r$ approaches infi…

Fang–Xia conjecture. Except for a finite number of cases, every finite non-abelian simple group has a cubic GRR.

Babai–Godsil conjecture. If $R$ is not generalised dicyclic or abelian of exponent greater than $2$, then for almost all inverse-closed subsets $S$ of $R$, $\mathop{\Gamma}(R,S)$ i…

Let $R$ be a finite group with a subgroup $M$ of index $2$. A bipartite DRR is a Cayley graph $\operatorname{Cay}(R,S)$, with $S\subseteq R\setminus M$, whose automorphism group is…

Watkins' conjecture. There is an integer $n$ such that every group of cardinality at least $n$ which is neither abelian of exponent greater than $2$ nor generalized dicyclic admits…

Spiga's conjecture. Except for a finite number of cases and for the groups $\operatorname{PSL}2(q)$, every finite nonabelian simple group has a cubic GRR whose connection set conta…

Xia–Fang finiteness conjecture. There are only finitely many finite nonabelian simple groups that have no cubic GRR.

Let $G$ range over finite nonabelian simple groups, and call $G$ cubic-GRR-free if it has no cubic graphical regular representation. Finiteness conjecture. There are only finitely…

Let $G$ be a finite nonabelian simple group. A cubic graphical regular representation (GRR) of $G$ is a Cayley graph $\mathrm{Cay}(G,S)$ of valency three whose automorphism group i…

For fixed $d\geq 3$, let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$, and let $GRR_d(n)$ count those that are graphical regular…