3 problems

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 planar digraph with digirth at least $g$, where $g\geq 3$, and let $\vec{\chi}^\ast(D)$ denote its star dichromatic number. Generalized 2-colour conjecture. … Equivale…