10 problems

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…

Non-concavity conjecture. There exists $\delta=\delta(k,d)>0$ such that $p\mapsto q_c(p)$ is not concave on $[0,\delta]$.

Extension conjecture. There is an absolute constant $C$ such that, for every integer $k$, every $k$-extension of an oriented tree of order $n$ is $C^k(2n-2)$-unavoidable; that is,

Conjecture on converse-invariant tree orientations. $D$ is converse invariant if and only if

Let $\Delta\in\mathbb N$, and let $c$ be a constant. The bounded-degree spanning-tree conjecture. For every $\Delta$ there is a $c$ such that every sufficiently large $n$-vertex di…

Let $k\in\mathbb N$, let $\ell\in\mathbb N$, let $D$ be a digraph, and let $T$ be an oriented tree with $k$ edges and $\ell$ leaves. Havet's conjecture. Every $(k+\ell)$-chromatic…

For each integer $k$, let $g(k)$ be the smallest integer such that every tournament on $n+g(k)$ vertices contains a copy of every oriented tree on $n$ vertices with $k$ leaves. Dro…

Let a $\delta^+$-maderian digraph be a digraph contained as a subdivision in every digraph whose minimum out-degree is sufficiently large. An oriented tree is an orientation of an…

Balanced binary tree unavoidability conjecture. $B_d$ is unavoidable for sufficiently large $d$, possibly with $d>1$ sufficient.

Bender and Wormald's conjecture. Almost all labelled oriented trees are unavoidable; equivalently, for every tournament $G$ on $n$ vertices, almost all labelled oriented trees on…