24 problems

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…

A graph pair $(\Gamma,\Sigma)$ is nontrivial if $\Gamma$ and $\Sigma$ are coprime connected twin-free graphs and exactly one of them is bipartite. A graph pair is stable if it has…

Five-color conjecture. Every such graph $G$ has a majority distinguishing edge $5$-coloring.

Let $n\geq 9$ be a composite integer. A nut graph is a graph whose adjacency matrix has nullity one and whose nullspace is spanned by a vector with no zero entries; write $o_v(G)$…

Wilson's conjecture. Every non-trivially unstable Rose Window graph is isomorphic to a graph in one of the families W1--W9.

Let $M$ be a model of set theory satisfying the Axiom of Choice, and let $M'=M/\mathcal{F}$ be a permutation model inside $M$. Suppose that Frucht's Theorem fails in $M'$. An inter…

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 v…

Strengthened orbit-interval conjecture. Then $\mathcal{O}=[e,v(w)]$. This is presented as a strengthening of the orbit-interval conjecture; its status is not resolved in the suppli…

Let $w \in \mathfrak{S}_n$ and let $\mathcal{O}=\\{\varphi(e) \mid \varphi \in \operatorname{Aut}(\Gamma(e,w))\\}$ be the orbit of the identity under graph automorphisms of…

Let $S$ be a surface, and let an object be naturally associated to $S$ and have sufficiently rich structure. Ivanov's metaconjecture. Every such object has the extended mapping cla…

Asymmetric fractalizer conjecture. There exists an asymmetric fractalizer on at most $9$ vertices.

Let $X$ be a circulant, that is, a Cayley graph on a cyclic group. Its canonical double cover $B(X)$ is the bipartite graph with vertex set $V(X)\times\mathbb{Z}_2$ in which…

Let $G$ be a connected, locally finite graph. The graph $G$ has infinite motion if every non-identity automorphism moves infinitely many vertices. A vertex colouring is asymmetric…

Let $G$ be a connected, finite, regular graph, and let $D'(G)$ denote the least number of colours needed to colour the edges of $G$ so that the only colour-preserving automorphism…

Let $G$ and $H$ be $2$-connected graphs, and let $f: V(G) \to V(H)$ be any mapping. The group $\tilde A(G,H,f)$ and the automorphism group of the Sierpiński product $G \otimes_f H$…

Let $S_n$ be the symmetric group on $n$ letters, and let $\operatorname{fix}(S_n)$ denote the set of possible numbers of vertices that fix a graph whose automorphism group is $S_n$…

Let $\Gamma$ be a graph. Its canonical double cover has adjacency matrix $\begin{pmatrix}0&A\A&0\end{pmatrix}$ when $A$ is the adjacency matrix of $\Gamma$. The matrix $A$ is comp…

The ORR classification conjecture. Every finite group $G$ admits an ORR, unless one of the following occurs:

Let $G$ be a $k$-regular graph, meaning that every vertex of $G$ has degree $k$. The distinguishing index of $G$, denoted $D'(G)$, is the least number of edge labels needed so that…

Let $\mathbb G$ be a $d$-dimensional transitive graph for some $d>8$, let $\mathfrak F$ be its uniform spanning forest, and let $r\geq 1$. Write $\mathcal C_r(\mathfrak F)$ for the…

Let $G$ be a countable, connected graph. An automorphism moves infinitely many edges if it does not fix infinitely many edges of $G$. A $2$-edge colouring is distinguishing if the…

Let $G$ be a graph, and write $G^2$ for its square, with $Aut(G)$ and $Aut(G^2)$ denoting their respective automorphism groups. For a connected graph, let $d$ be its diameter and…

Unbounded separation conjecture. For any $N>0$, there exists a graph $\Gamma$ and automorphism group $G$ of $\Gamma$ for which $OP_{\Gamma,G}(x)$ has a root at least $N$ larger tha…

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 $n\to\infty$.