34 problems

Abreu et al.'s conjecture. $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs.

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 $D$ be a finite subset of the positive integers such that $1\in D$. For a numerical semigroup $S$, let $\Delta_0(S)$ and $\Delta_\infty(S)$ denote its 0-delta set and infinity-…

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 $[m]=\{1,2,\dots,m\}$, 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…

Let $\Sigma$ be an alphabet with a reflexive and symmetric commutation relation $\vartheta$, and let ${\rm \bf M}(\Sigma,\vartheta)$ be the associated trace monoid. Consider the tr…

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 $n\to\infty$,

Chor–Lemke–Mador conjecture. There exists a positive constant $C$ such that

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.…

Let $K_{2n}$ be the complete graph on $2n$ vertices, and let $L$ be a random $(p,2n-1)$-list assignment, in which each colour from $[2n-1]$ is included independently in each edge's…

Let $G$ be a complex reflection group and let $g\in G$. For a minimum-length reflection factorization $f$ of $g$, write $G_f$ for the subgroup generated by its factors, and conside…

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 $M$ 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 $A$ be an alphabet, and let $A^\omega$ denote the set of infinite words over $A$. A generalized Lyndon word is a finite or infinite word that is Lyndon with respect to a genera…

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…