126 problems

Central-quotient eigengap conjecture. Either $k_{>1}=|G|-1$, or

Let $s\geq 3$ be an integer. An $s$-spanning set of size $2$ in a finite abelian group is a set of two elements whose signed sums of at most $s$ terms cover the group. Largest nonc…

Uniform expansion conjecture. If $G$ is $\varepsilon$-quasirandom, then all connected Cayley graphs of $G$ are $\varepsilon$-uniform expanders.

Xu's conjecture. The proportion of inverse-closed subsets $S$ of $R$ such that $\operatorname{Cay}(R,S)$ is a normal Cayley graph of $R$ approaches $1$ as $r$ approaches infinity.

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…

Let $G=\operatorname{Sym}_n$ and let $T$ be the set of transpositions, so that $\operatorname{Sym}_n(T)$ is the corresponding Cayley graph, $B_r(\sigma)$ denotes the radius-$r$ bal…

Let $(G_i)$ be the family of all non-abelian finite simple groups. A Cayley graph $\mathcal{C}(G_i,S_i)$ is the graph associated with the generating set $S_i$, and a family of grap…

Let $\Gamma_i$ be a family of finite groups, and let $\Sigma_i$ and $\Sigma_i'$ be generating sets of $\Gamma_i$ whose sizes are uniformly bounded. Lubotzky–Weiss generating-set co…

Let $X_n$ be the unitary Cayley graph considered in the paper, and let $m(r)$ denote the maximum length of an induced cycle in $X_n$ when the underlying modulus has $r$ distinct pr…

Bedert–Drăganić–Müyesser–Pavez-Signé conjecture. There is an absolute constant $C$ such that, for every connected Cayley graph $G$ of order $n$ and degree $d$, the condition

Rapaport–Strasser conjecture. Every connected Cayley graph on a finite group with at least three elements has a Hamilton cycle.

Let $G$ be a finite group, let $S$ be a generating set, let $n=|G|$, and let $\mathfrak{D}(\operatorname{Cay}(G,S))$ and $\mathcal{A}(\operatorname{Cay}(G,S))$ denote the graph dia…

Let $G$ be a finite group, let $S$ be a generating set, let $n$ denote the group order, let $g_{\mathrm{min}}$ be the minimal number of generators of $G$, and let…

Let $G$ be a perfect group and let $S$ be a generating set subject to the restrictions required for the square clustering coefficient. Write $\square(\operatorname{Cay}(G,S))$ for…

Cashman–Kelley’s conjecture. For $p>2\ell+1$, the Cayley–Turán number of the odd cycle $C_{2\ell+1}$ satisfies

Let $n$ and $k$ be positive integers. Consider the Cayley graph on $4\mathds{Z}/n\mathds{Z}4$ with generating set $\{-k,\ldots,k\}\setminus\{0\}$. Espuny Díaz, Lichev, and Wesolek'…

Cayley classification conjecture. Every Seymour Cayley orientation can be constructed by taking, possibly repeatedly, lexicographic products of empty graphs, powers of directed cyc…

Let $G$ be a Cayley graph on $n$ vertices with degree $d$, and let $G_p$ be the random graph obtained by retaining each edge of $G$ independently with probability $p$. Random Cayle…

Given a finite group $G$ and a symmetric subset $S\subset G$, the Cayley graph $\operatorname{Cay}_G(S)$ has vertex set $G$ and edges $\{g,gs\}$ for $g\in G$ and $s\in S$. A graph…

Stability conjecture. For a group $G$ of order $r$, the proportion of inverse-closed subsets $S$ of $G$ such that $\operatorname{Cay}(G,S)$ is stable approaches $1$ as $r$ tends to…

Let $G$ be a finite group. A Cayley graph of $G$ is formed from a symmetric subset $S\subset G$ by joining $x,y\in G$ when $x^{-1}y\in S$. Such a graph is $C$-Ramsey if it has no c…

Let $r$ be a positive integer. Let $M_X$ be an $m\times r$ integer matrix and let $X=M_X^{SACG}$ denote the Abelian Cayley graph associated with $M_X$. Assume that $M_X$ has no zer…

Let $\Gamma$ be a finite abelian group with generating set $S$, and let $G=\operatorname{Cay}(\Gamma,S)$. Let $\mathcal{H}$ denote the class of graphs having a unique Hamiltonian c…

Let $S_n$ be the symmetric group on $[n]:=\{1,\dots,n\}$, let $T_n$ be the set of transpositions in $S_n$, and let $Cay(S_n,T_n)$ be the Cayley graph of $S_n$ generated by $T_n$, w…

Higher-residue conjecture. As $N\to\infty$ through the primes congruent to $1$ modulo $r$,