12 problems

Spiga's element-and-involution conjecture. Except for $\mathrm{PSL}_2(q)$ and a finite number of other cases, every finite non-abelian simple group $G$ contains an element $x$ and…

Fang–Li–Wang–Xu conjecture. Every finite non-abelian simple group admits a cubic GRR.

Spiga's asymptotic conjecture. As the order of $G$ tends to infinity, the proportion of cubic Cayley graphs of $G$ that are GRRs approaches $1$.

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

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…

Let $R$ be a finite group with a subgroup $M$ of index $2$. A bipartite GRR for $(R,M)$ is a Cayley graph of the form $\operatorname{Cay}(R,S)$, where $S$ is an inverse-closed subs…

Watkins' conjecture. Being Class II is the only obstruction to admitting a graphical regular representation: finite groups split into groups admitting a GRR and Class II groups.

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…

Watkins' finiteness conjecture. Apart from these two infinite families, only finitely many groups fail to admit a GRR, and all such exceptional groups are finite.

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…

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…