116 problems

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Burr's conjecture. Every oriented tree of order $n$ is $(2n-2)$-universal.

Magnant–Martin's strengthened conjecture. If $G$ is a $d$-regular digraph on $n$ vertices, then

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…

An oriented graph $G$ on $n$ vertices has an edge for each oriented edge of its underlying graph. An antidirected tree is an orientation of a tree in which every vertex has either…

Häggkvist's conjecture. If

Let $d_k(n)$ be the minimum number of arcs in a $k$-dicritical digraph of order $n$, and let $o_k(n)$ be the minimum number of arcs in a $k$-dicritical oriented graph of order $n$,…

Jackson's conjecture. For each $d>2$, every $d$-regular oriented graph on $n\leq 4d+1$ vertices has a Hamilton cycle.

Neumann–Lara's conjecture. Every orientation of a planar graph has dichromatic number at most $2$.

McDiarmid–Mohar conjecture. Every oriented graph $D$ satisfies

Belkhechine et al.'s inversion-number conjecture.

Let $\vv{Q_d}$ denote the oriented $d$-dimensional hypercube, and let $\vv{r}(\vv{Q_d})$ be its oriented Ramsey number. Directed Burr–Erdős conjecture. There is an absolute constan…

Let $G$ be an oriented graph, let $\b7$ denote its minimum semidegree, and let $k$ be a nonnegative integer. An orientation of the $k$-edge path is an oriented graph obtained by as…

Harutyunyan–Mohar conjecture. Every oriented graph $D$ satisfies

Häggkvist's conjecture. Every oriented graph $D$ with

A $T_3$-free oriented graph is an oriented graph containing no transitive tournament on three vertices. Cherlin's conjecture. Almost all $T_3$-free oriented graphs are tripartite.…

Let $G$ be an oriented graph, meaning a directed graph with no 2-cycles, on $n$ vertices. Let $\delta^0(G)$ denote its minimum semi-degree, the minimum of its minimum in-degree and…

Asymptotic coloring conjecture. As $s\rightarrow\infty$,

Let $k\in\mathbb{N}$ and let $D_1,\dots,D_k$ be oriented graphs satisfying … Write $[D_1,\dots,D_k]$ for their join, and let $\operatorname{inv}$ denote inversion number. Dijoin co…

Vertex-deletion conjecture. There exists $v\in V(D)$ such that

Dijoin counterexample-range conjecture. For all $\ell,r\in\mathbb{N}$ with $\ell\geq 3$ or $r\geq 3$, there exist oriented graphs $L$ and $R$ such that

Dijoin conjecture.

The all-$k$ dijoin counterexample conjecture. For any $k\geq3$, there is a tournament $T_k$ with $\operatorname{inv}(T_k)=k$ such that

Alon's inversion-number partition conjecture. For every two positive integers $k_1,k_2$, there exists an integer $f(k_1,k_2)$ such that every oriented graph $D$ with

Bang-Jensen et al.'s complexity conjecture. Deciding whether a given digraph $D$ has inversion number at most $k$ is NP-complete for any fixed positive integer $k$.