17 problems

El-Zahar's conjecture. If

Asymptotic excess conjecture. As $d\to\infty$,

Christoph–Draganić–Girão–Hurley–Michel–Müyesser conjecture. If $d\mid n$, then the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph on…

Cycle-factor extension conjecture. If

The $C_4$-factor conjecture. There exists $n_0=n_0(\eta)\in\mathbb N$ such that, for every $n\geq n_0$ divisible by $4$, if

Let $G$ be a spanning subgraph of the $n$-blow-up of $C_k$, where $k\geq 4$, and let $\delta^*(G)$ denote its relative minimum degree. A transversal $C_k$-factor is a collection of…

Let $D$ be a digraph of order $n$, with minimum semi-degree $\delta^0(D)$. Let $n=n_1+\cdots+n_k$ be a positive integer partition with $n_i\geq 3$ for each $i$. A cycle factor of t…

Let $G$ be a graph of order $n$, let $k$ be a positive integer, let $e_1,\ldots,e_k$ be $k$ disjoint edges in $G$, and let $n=n_1+\cdots+n_k$ be an integer partition with…

Let $D$ be a $k$-fully regular $c$-partite tournament with $c\geq5$, meaning that for every pair of distinct partite sets, their induced bipartite tournament is $k$-regular. A…

Let $D$ be a regular $c$-partite tournament, with $c\geq 5$. A $(p,|V(D)|-p)$-cycle-factor is a cycle-factor consisting of cycles of lengths $p$ and $|V(D)|-p$. Yeo's conjecture. F…

Let $D$ be a $k$-regular bipartite tournament, where $k$ is an integer greater than $2$, and let $u$ and $v$ be two specified vertices of $D$. Let $F_{4k}$ and the other specified…

Let $\mathcal{Q}^n$ be the $n$-dimensional hypercube, and let $\mathcal{Q}^n_p$ be the random subgraph obtained by retaining each edge independently with probability $p$. For fixed…

Wang's conjecture. If $\delta(W)\geq (3n-3)/2$, then $D$ contains an arbitrary $W$-cycle-factor.

Wang's conjecture. If $|W|\geq 3k$ and $\delta(W)\geq 2n/3$, then $G$ contains an arbitrary $W$-cycle-factor.

Let $T$ be a tournament, and for $t\in\mathbb{N}$ let $f(t)$ be the least integer such that all strongly $f(t)$-connected tournaments, up to finitely many exceptions, contain all p…

Hamilton–Waterloo existence conjecture. If $k\equiv 2\pmod 4$ and $s>0$, then an $HW(n;r,s;n,k)$ exists if and only if

Cuckler and Yuster's conjecture. Every regular tournament on $n=6k+3$ vertices contains a triangle factor.