17 problems

An oriented planar graph is a planar digraph with no digons, and its dichromatic number is the least number of colours in a vertex colouring in which every colour class induces an…

Let $D_{3}^{n}$ be a planar digraph with $n$ vertices and directed girth $3$, and let ${\rm fvs}(D_{3}^{n})$ denote its minimum feedback vertex set size. Harutyunyan's conjecture.…

Woodall's conjecture. For every planar digraph $G$ with at least one dicycle,

Neumann–Lara's conjecture. Every planar digraph of digirth at least $3$ can be vertex-partitioned into two acyclic sets.

Goemans–Williamson's conjecture.

Low-density normal-set conjecture. Every plane digraph $G$ admits a normal set of cycles with low density and of size $fvs(G)$.

The minimum-energy conjecture.

Woodall's conjecture. There exists a packing of $g$ disjoint feedback arc sets in $D$.

Star-dichromatic conjecture.

Planar packing conjecture. There exists a packing of $g-1$ disjoint feedback vertex sets in $D$.

Higher-girth packing conjecture. There exists a packing of $g$ pairwise disjoint feedback vertex sets in $D$. Equivalently, $D$ can be vertex $g$-coloured such that every directed…

Two-Colour-Conjecture. Every oriented planar digraph $D$ is $2$-colourable.

Neumann-Lara's conjecture. Every loopless planar digraph admits an NL-$2$-coflow.

Planar digraph circular-colouring conjecture. Every planar digraph $D$ of directed girth at least $k\geq 3$ admits a circular homomorphism to $\vec{C}_{k-1}$. Equivalently,

2-dichromatic planar digraph conjecture. Every simple planar digraph is 2-dichromatic.

Let $D$ be a simple planar digraph on $n$ vertices. An acyclic set is a vertex set inducing no directed cycle in $D$. Hefetz's conjecture. Every simple $n$-vertex planar digraph ha…

Let $D$ be a simple planar digraph. Say that $D$ has no cycles of lengths $4,\dots,k$ when it contains no directed cycles whose lengths are among those integers. Gap-cycle conjectu…