211 problems

Lower-bound conjecture for tournament dijoins. One should have

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

Jackson's conjecture. Every regular bipartite 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…

Kühn–Osthus conjecture. Every regular tripartite tournament has a Hamilton decomposition.

Pokrovskiy's conjecture. For every $k\in\mathbb{N}$, there exists an integer $h(k)$ such that every $2k$-connected tournament with $\delta^0(D)\geq h(k)$ is $k$-linked.

Let $T$ be an oriented tree with $k$ edges and $ll$ leaves, and let a tournament be an orientation of a complete graph. Havet–Thomassé's conjecture. Every tournament on $k+\ell$ ve…

Belkhechine et al.'s $Q_n$ conjecture.

Let $\vec{H}$ be a strongly connected tournament with $k:=v(H)\geqslant 4$, and let $p\gg n^{-2/(k+1)}$. Tournament orientation-counting conjecture. Then, with high probability, ……

Directed Erdős–Hajnal conjecture. For every tournament $H$, there exists $epsilon(H)>0$ such that every $H$-free tournament with $n$ vertices contains a transitive subtournament of…

Let $k$ be a positive integer. A tournament is a directed graph in which exactly one of $xy$ and $yx$ is an edge for every pair of distinct vertices $x,y$. A tournament is $k$-link…

A digraph is semicomplete if it has no pair of non-adjacent vertices. A tournament is an orientation of a complete graph, hence a semicomplete digraph with no directed 2-cycles. A…

Pokrovskiy's conjecture. There exists an integer $f(k)$ such that every $2k$-strong tournament with minimum semi-degree at least $f(k)$ is $k$-linked.

For a tournament $T$ on $k$ vertices, let $g(k)$ be the largest integer such that every such tournament contains an acyclic subgraph with chromatic number $g(k)$. Nassar–Yuster con…

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…

Let $G$ be a regular tournament on $n$ vertices, and let $B(G)$ denote its brushing number. Regular-tournament brushing conjecture. If $G$ is regular, then … This extends the estab…

Approximation conjecture. There is a function $f$ such that for every integer $k$, there is a polynomial-time algorithm which, given a tournament $T$, correctly concludes that…

A tournament $R$ is a rebel if the class of tournaments not containing $R$ has bounded domination number. A poset tournament is a tournament admitting an ordering whose backedge gr…

Neumann-Lara's conjecture. The smallest $5$-chromatic tournament has order $17$; equivalently, $n_5=17$.

Linial–Morgenstern conjecture. For every tournament $T$, one has

Conjecture on the possible bases.

Let $cmathcal{P}$ be a hereditary property of tournaments, and let $cmathcal{P}_n$ denote the tournaments in $cmathcal{P}$ on $n$ vertices. There exists a constant $calpha>0$ and a…

Let $d$ be an odd positive integer, let $R_d$ be the cyclic regular tournament described above, and let $\mathcal{H}$ denote the Hessian quantity arising in the smooth-point asympt…

A bipartite Eulerian tournament is an orientation of a complete bipartite graph in which every vertex has equal indegree and outdegree. Let $C_4$ denote the directed cycle on four…

Exact transitive-triple packing conjecture.