5 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…

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…

Two-arborescence conjecture. If