10 problems

Let $H=(V,E,A)$ be a mixed graph, let $\mathcal{D}(V)$ denote the relevant family of subpartitions, and let $d_A^-(X)$ count arcs entering $X$. Let $k\geq0$ and $d\geq1$ be integer…

Frank's conjecture. The digraph $D$ is $r$-rooted $k$-vertex-connected if and only if it contains $k$ independent spanning $r$-arborescences.

Let a path of double-arborescences have $n$ vertices and clique number $k$. Minimum-word-representant length conjecture. The length of a minimum-word-representant of the path is al…

Equivalent Rainbow Arborescence Conjecture. If $k \ge n-1$, then the disjoint union $G$ of $k$ spanning arborescences has a rainbow spanning arborescence $B$.

Yokoi's Rainbow Arborescence Conjecture. If $G$ is the disjoint union of $n-1$ spanning arborescences $A_1, A_2, \dots, A_{n-1}$, then $G$ has a rainbow spanning arborescence $B$.

Let $G$ be an Eulerian planar graph. An orientation is alternatingly in- and outward oriented at each vertex when, in the cyclic order around every vertex, its incident edges alter…

Let $K_n$ be the complete graph on an odd number $n$ of vertices, and let ${\rm allarb}(O)$ denote the total number of arborescences of an orientation $O$ of $K_n$. An Eulerian ori…

Maker's arborescence cost conjecture. W.h.p. over the random choice of $C$, Maker can construct a spanning arborescence $T$ satisfying

Almost Spanning Two Rooted-Arborescences conjecture (ASTRA). The digraph $D$ admits an out-arborescence and an in-arborescence that are edge-disjoint, have the same root, and each…

Two-arborescence conjecture. If