165 problems

Let $T$ be an oriented tree on $n$ vertices. An oriented graph $H$ is $\chi$-universal if every digraph of chromatic number $\chi$ contains $H$ as a subdigraph. Burr's conjecture.…

Let $T$ be an oriented tree on $n$ vertices, and let $\vv{r}(T)$ be the least $N$ such that every tournament on $N$ vertices contains a copy of $T$. Sumner's conjecture. … This is…

Häggkvist's conjecture. If

Kelly's conjecture. Every regular tournament has a Hamilton decomposition.

A tournament is an oriented complete graph. An oriented Hamilton path is an oriented path containing every vertex of the tournament exactly once. Rosenfeld's conjecture. Every tour…

Let $P$ be an oriented path with two blocks on $n\ge 4$ vertices. An oriented graph $H$ is $\chi$-universal if every digraph of chromatic number $\chi$ contains $H$ as a subdigraph…

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…

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 $n$,…

An oriented graph is a digraph with at most one arc between any two vertices. The transitive tournament $T_k$ is the orientation of the complete graph $K_k$ whose vertices can be l…

Let $\overrightarrow{F}$ be an antidirected tree with $k$ arcs, meaning that every vertex of $\overrightarrow{F}$ is either a source or a sink. Write…

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…

Treglown's conjecture. There exists $n_0$ such that, for every $n\in 3\mathbb{Z}$ with $n\geq n_0$, every oriented graph $G$ on $n$ vertices satisfying

Let $G$ be an oriented graph, and let $4\delta^0(G)4$ denote its minimum semidegree, the minimum of the in-degree and out-degree over all vertices. An oriented path of length…

Häggkvist–Thomason conjecture. If

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

Let $G$ be an oriented graph. Its minimum pseudo-semidegree $\bar{\delta}^0(G)$ is the minimum among all non-zero in-degrees and out-degrees of its vertices; equivalently, it is th…

Let $D$ be an oriented graph, and let $\Delta$ be the maximum degree of the underlying graph of $D$. Let $\vec{\chi}(D)$ denote its dichromatic number. Erdős–Neumann-Lara conjectur…

Let $G$ be an oriented graph on $n$ vertices, and let $\delta(G)$ denote its minimum degree. A Hamilton cycle has an edge in one of the two possible directions for each consecutive…

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…

Let $f(G)$ denote the oriented diameter parameter for a graph $G$, and let ${\sf K}(2,p,q)$ be the complete tripartite graph with part sizes $2$, $p$, and $q$, where $p$ and $q$ ar…

A tournament is an orientation of a complete graph. For an oriented graph $H$, its oriented Ramsey number, denoted by $\overrightarrow{r}(H)$, is the smallest integer $N$ such that…

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

Let $D$ be a $k$-dicritical oriented graph on $n$ vertices, and let $m(D)$ denote its number of arcs. Hoshino–Kawarabayashi conjecture. For every integer $k\geq3$, … Hoshino and Ka…

Let $G$ be a graph. An orientation of $G$ is consecutively colourable if it has a proper arc colouring such that, for every vertex $v$, the colours of all out-arcs from $v$ and the…