14 problems

Let $G$ be a group, and let $\Sigma_d(G)$ be the graph whose vertices are the ordered generating $d$-tuples of $G$, with two vertices adjacent if and only if they differ in exactly…

Let $G$ be a finite group, and let $\bar\Delta(G)$ denote its generating graph, whose vertices represent the elements of $G$ and whose edges join pairs that generate $G$. Generatin…

Induced 5-hole conjecture. There exists a subset $X$ of $G$ such that the subgraph of $\Gamma(G)$ induced by $X$ is a 5-hole.

Non-perfectness conjecture. The generating graph $\Gamma(G)$ is not perfect.

Let $G$ be a finite group. The generating graph $\Gamma(G)$ has vertex set $G$, with two distinct vertices adjacent when they generate $G$. A vertex is isolated if it has no adjace…

Let $G$ be a finite group, let $d(G)$ be the smallest size of a generating set for $G$, and let $k\geqslant d(G)$ be an integer. The product replacement graph $\Gamma_k(G)$ has as…

Brittenham–Lucchini–Maróti–Magaard conjecture. If $\Gamma(G)$ has at most one isolated vertex, necessarily the identity, then $\Delta(G)$ is Hamiltonian.

Let $G=\mathbb{Z}\times\mathbb{Z}$, and let $\Gamma(G)$ be the graph whose vertices are the elements of $G$, with two vertices adjacent when they generate $G$. Let $\Gamma^*(G)$ be…

Let $G$ be a finite group with $|G|\geq 4$, and let $\Gamma(G)$ be its generating graph. Equivalence conjecture. The following conditions are equivalent: $G$ has spread $1$; $G$ ha…

Let $G$ be a finite group with $|G|\geq 4$, and let $\Gamma(G)$ be its generating graph: the vertices are the non-identity elements, and $x$ and $y$ are adjacent exactly when…

Let $G$ be a finite group with $|G|\geqslant 4$, and let $\Gamma(G)$ be its generating graph. The spread of $G$ is the largest integer $k$ such that, for every set of $k$ nonidenti…

Nonzero-spread bound conjecture.

Nonzero-spread equivalence conjecture. If $G$ has nonzero spread, then

One-stabilization conjecture. The tuples $(t_1,\ldots,t_n,1)$ and $(s_1,\ldots,s_n,1)$ are connected in $\bar X_{n+1}(G)$.