35 problems

Rainbow minimum-threshold conjecture. There exists $n_0$ such that, for every $n\geq n_0$, if

DeBiasio's conjecture. Every $n$-vertex digraph $D$ satisfying

An oriented graph is a directed graph with at most one directed edge between any pair of vertices. For an oriented graph $G$, let $V(G)$ be its vertex set, let $n=|V(G)|$, and writ…

Häggkvist's conjecture. If

A digraph $D$ has order $n\ge 3$. A pair of nonadjacent vertices is dominated if it has a common in-neighbor, and dominating if it has a common out-neighbor. Let $d(u)$ denote the…

Let $G$ be a claw-free graph, meaning that $G$ has no induced subgraph isomorphic to $K_{1,3}$. Let $\delta(G)$ denote its minimum degree and let $\alpha(G)$ denote its independenc…

Let $D$ be an oriented graph on $n\ge 3$ vertices. For distinct vertices $u,v$, call them non-adjacent if neither directed arc joins them, and define … when $D$ has a pair of non-a…

Let $D$ be an oriented graph on $n\ge 3$ vertices with minimum degree $\delta$. For vertices $u,v$, write $d(u)$ and $d(v)$ for their degrees, and call them non-adjacent if neither…

Let $H$ be a 3-graph of order $n$, let $\sigma_2(H)$ denote the minimum $\deg(u)+\deg(v)$ over adjacent vertices $u,v$, and let $H^{1,2}_{n,x,y}$ be the construction defined in the so…

Let $H$ be a 3-graph of order $n$, let $\sigma_2(H)$ be the minimum of $\deg(u)+\deg(v)$ over adjacent vertices $u,v$, and let $H_{n,s}^2$ be the corresponding extremal 3-graph. Zhang…

Let $G$ be a 2-connected graph on $n$ vertices, and let $x,y\in V(G)$. Let $0<\alpha\leq\frac{1}{2}$. Alpha-generalized path conjecture. If $G-\{x,y\}$ contains more than…

Let $G$ be a 2-connected graph on $n$ vertices. Häggkvist–Jackson conjecture. If $G$ contains at least … vertices of degree at least $k$, then $G$ has a cycle of length at least ……

Let $G$) be a 2-connected graph on $n$ vertices, and let $c(G)$ denote the length of a longest cycle in $G$. Woodall's conjecture. If there are at least $\frac{n}{2}+k$ vertices of…

The $2n/3$ minimum semi-degree conjecture. If

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

Wang's conjecture. If $\delta^0(D)\geq (9k-3)/4$, then $D$ contains $k$ vertex-disjoint directed cycles, each of order at least $3$.

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

Bang-Jensen et al.'s conjecture. If

Let $k\in\mathbb{N}$, and let $G$ be a graph. For a vertex $x\in V(G)$, let $N(x)$ denote its neighbourhood and let $N_2(x)$ denote its second neighbourhood. Large first- and secon…

Let $k\in\mathbb{N}$, let $0\leq\alpha\leq \frac{1}{2}$, and let $G$ be a graph. Write $\delta(G)$ and $\Delta(G)$ for the minimum and maximum degrees of $G$, respectively. Interpo…

Let $k,\Delta\in\mathbb{N}$, and let $G$ be a graph. Write $\delta(G)$ and $\Delta(G)$ for the minimum and maximum degrees of $G$, respectively. Constant-degree-tree conjecture. If…

Let $H$ be a simple $3$-partite hypergraph with sides $A$, $B$, and $C$, where simplicity means that no edge is repeated. Let $\nu(H)$ denote the maximum matching size. Generalized…

Let $H$ be a simple $3$-partite hypergraph with sides $A$, $B$, and $C$. Write $\delta(A)$ for the minimum degree of a vertex in $A$, $\Delta(B\cup C)$ for the maximum degree of a…

Smaller matching threshold conjecture. Given $1\leq d\leq k-2$, there exist $n_0$ and $C$ such that

Bang-Jensen–Gutin–Li conjecture. If