124 problems

Let $n$ be sufficiently large, and let $G$ be a regular tripartite tournament on $3n$ vertices. Denote by ${\mathcal T}_\triangle$ the known regular tripartite tournament obtained…

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

The $Q_n$ inversion-number conjecture.

Price-of-symmetrisation conjecture. For $2\leq n\leq 10$,

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…

Exact transitive-triple packing conjecture.

Let $T$ be a tournament, namely an oriented graph in which every pair of distinct vertices is joined by exactly one directed arc. For a vertex $x$, let $N_T^+(x)$ and $N_T^{++}(x)$…

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…

Bang-Jensen–DeVos–Mütze conjecture. Every $(2k-1)$-strong semicomplete digraph on at least $2k+1$ vertices contains a spanning $k$-strong tournament.

Let $Q_n$ be the tournament on $[n]=\{1,\ldots,n\}$ obtained from the natural transitive tournament by reversing precisely the consecutive pairs $12,23,\ldots,(n-1)n$. Equivalently…

Let $T$ be an orientation of a tree with maximum degree at least $3$. An oriented graph $D$ is converse invariant if $f_T(D)=f_T(\overline{D})$ for every tournament $T$, where…

Deficient-sequence characterization conjecture. A partition $\lambda$ is complete if and only if, for every $\lambda'\subseteq\lambda$, the set $\mathfrak{S}^2(\lambda')$ has no de…

Complete-partition classification conjecture. A partition $\lambda$ is complete if and only if

Bebeacua's conjecture. For every positive integer $n$,

A poset tournament is a tournament $T$ for which there exists a total ordering $<$ of $V(T)$ such that, for all $u<v<w$, if $u\to_T v$ and $v\to_T w$, then $u\to_T w$. Equivalently…

An explicit family of tournaments is a sequence $(T_n)_{n\geq 1}$ of tournaments, with $|V(T_n)|$ denoting the number of vertices and $\vec\omega(T_n)$ denoting the clique number.…

Let $\mathcal{C}$ be a class of tournaments, where a tournament is an orientation of a complete graph. Let $\mathcal{C}^{subst}$ denote the closure of $\mathcal{C}$ under substitut…

Let $\mathcal{G}(n,k)$ be the probability space of $k$-majority tournaments on vertex set $[n]$ obtained by uniformly choosing, with replacement, $2k-1$ linear orders of $[n]$. Let…

Let $D$ be an oriented graph with $n$ vertices and $m$ edges or arcs, and let $\vec{\alpha}(D)$ denote its largest acyclic set. Aharoni–Berger–Kfir conjecture. … For tournaments th…

Let $T_1,T_2,\ldots$ be tournaments, and write $V(T_n)$ for the vertex set of $T_n$. Explicit tournament construction conjecture. There is an explicit construction of tournaments…

A tournament is a directed graph in which exactly one of the two possible directions is chosen for every pair of distinct vertices. For a tournament $T$, its directed clique number…

Let a link-irregular tournament be a tournament in which the directed links of every pair of distinct vertices are non-isomorphic. Link-irregular tournament conjecture. A link-irre…

Let $T$ be a strong tournament of order $n\geq 8$, let $e$ be an arc of $T$, and let $P$ be an oriented path of order $n$. El Sahili's conjecture. The tournament $T-e$ contains $P$…

Tournament acyclic dicolouring algorithm conjecture. For every fixed $k\geq3$, it is polynomial-time decidable whether a tournament $T$ satisfies