20 problems

Let $[m]=\{1,2,\dots,m\}$, let $\mu\vdash k$, and let $\omega_\mu$ be a fixed element of the conjugacy class $C_\mu$ in $\mathfrak{S}_k$. Let $\mathfrak{S}^{(m)}_k$ be the set of p…

For an even integer $n$, let $f(n)$ be the maximum integer $k$ such that some $k$-regular graph $G$ on $n$ vertices admits a flawless 1-factorization, meaning a perfect 1-factoriza…

Let $\mathrm{GL}^\varepsilon_n$ denote the general linear group when $\varepsilon=1$ and the general unitary group when $\varepsilon=-1$, and let $\mathrm{GL}^\varepsilon_1$ be its…

Abreu–Diwan–Jackson–Labbate–Schwenk's conjecture. $G$ must be $K_{3,3}$, the Heawood graph or the Pappus graph.

Abreu–Diwan–Jackson–Labbate–Schwenk's conjecture. $G$ is pseudo 2-factor isomorphic if and only if $G$ can be obtained from $K_{3,3}$, the Heawood graph or the Pappus graph by repe…

Minimal-order conjecture. As $n\to\infty$,

Let $S_n$ denote the symmetric group of degree $n$, and let “multifold-factorizable” have the meaning defined in the paper. Symmetric-group multifold-factorization conjecture. The…

Let $A_n$ denote the alternating group of degree $n$, and let “multifold-factorizable” have the meaning defined in the paper. Alternating-group multifold-factorization conjecture.…

A $2$-factor is a spanning $2$-regular subgraph, and a full rainbow matching is a matching containing one edge from every color class. Alspach's conjecture. If $G$ is a simple $2d$…

Let $m$, $n$, and $k$ be positive integers with $m\geq 2$, and suppose … For a prime $p$ dividing $|G|$, write $\operatorname{ord}_p(m)$ for the multiplicative order of $m$ modulo…

Let $k$, $\omega$, and $n$ be positive integers, and let $\mathbb{Z}_{\omega}$ be a finite cyclic group of order $\omega$. Suppose … A subset $A$ is a direct factor of…

Let $m$, $n$, and $k$ be positive integers with $m\geq 2$, and suppose … A multiplier set $M$ splits an abelian group $G$ if there is a set $S\subseteq G$ such that every nonzero e…

Let $W$ be a well-generated finite complex reflection group, and let $T$ and $T'$ be reflection factorizations of a Coxeter element of $W$. Two factorizations are said to have the…

Let $M$ be the generalized arithmetic monoid under consideration, let $m\in M$, and let $\mathcal Z(m)$ be its set of factorizations. For factorizations $z,z'\in\mathcal Z(m)$, wri…

Let $M$ be the generalized arithmetic monoid under consideration, with parameters $h$ and $d$, and let $c(M)$, $c_{mon}(M)$, and $c_{eq}(M)$ denote its regular, monotone, and equiv…

Pasotti–Pellegrini conjecture. There exists a near $1$-factor $F$ of $K_v$ with $\ell(F)=L$ if and only if $L$ is admissible.

Let $n$ and $m$ be positive integers such that $m=2n-2$, and let $r\geq 2$. Suppose that $r$, $\mu$, and $\lambda$ are positive integers independent of $n$, with $\mu>\lambda$. A d…

Let $W$ be a well-generated finite complex reflection group, let $c$ be a Coxeter element of $W$, and consider factorizations of $c$ into reflections. Two such reflection factoriza…

Let $G$ be a simple $3$-uniform hypergraph, let $M(G)$ be its adjacency matrix, let $\operatorname{per} M(G)$ denote the permanent of that matrix, and let $\varphi(G)$ denote the n…

Let $g$ be a regular elliptic element of $GL_n({\mathbb{F}}_q)$, and let $t_q(n,\ell)$ denote the number of ordered reflection factorizations of a Singer cycle of length $\ell$. Un…