6 problems

Let $G$ be a plane graph, and let $u$ be a positive integer. An $nu$-facial edge-coloring of $G$ is an edge-coloring in which all edges on every facial trail of length at most…

Let $G$ be a plane graph, and let $\ell$ be a positive integer. An $\ell$-facial vertex coloring of $G$ is a vertex coloring in which vertices occurring on the same facial walk wit…

Fabrici et al.'s conjecture. If $G$ is a $2$-edge-connected plane graph, then

Fabrici–Gö ring's conjecture. If $G$ is a plane graph, then there is a proper coloring of the vertices of $G$ by colors in $\{1,2,3,4\}$ such that every face contains a unique vert…

Let $G$ be a plane graph and let $\ell$ be a positive integer. An $\ell$-facial coloring is a vertex coloring such that any two vertices joined by a facial walk of length at most…

Let $G$ be a plane graph. An $\ell$-facial edge coloring of $G$ is an edge coloring in which any two edges at distance at most $\ell$ on a boundary walk of some face receive distin…