126 problems

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

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 $s\geq 3$ 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…

Let $G=\langle S\rangle$ be a two-ended group, and suppose that its commutator subgroup satisfies … The Hamiltonian circle conjecture. The Cayley graph … has a hamiltonian circle.…

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…

Generalized Petersen graph conjecture. Every generalized Petersen graph is an induced subgraph of a minimal Cayley graph.

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…

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.

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 $G$ 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 $G$ be a group, let $k$ satisfy $k \gg \log |G|$ and $\log k \ll \log |G|$, and let $\mathcal{G}_k$ be the Cayley graph of $G$ generated by $k$ independently and uniformly chos…

Let $G=\langle S\rangle$ be a one-ended generalized quasi-dihedral group, and let $\operatorname{Cay}(G,S)$ denote its Cayley graph. A Hamiltonian double ray is a spanning double r…

Let $G$ be a nonabelian finite simple group. A subset $S\subset G$ is symmetric when it is closed under taking inverses, and $\operatorname{Cay}(G,S)$ denotes the Cayley graph of…

Let $S_n$ be the symmetric group, and let $T_n$ be its transposition graph. For $\mathcal{A} \subset S_n$, write $\partial \mathcal{A}$ for its edge-boundary in $T_n$. Let…

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

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

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 $\Gamma_i$ 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 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'…