Borodin–Broersma–Glebov–van den Heuvel conjecture on cyclic coloring

Let $G$ be a plane graph, let $\Delta^*(G)$ be its maximum face degree, and let $k^*=k^*(G)$ denote the maximum number of vertices that two faces of $G$ can have in common. A cyclic coloring assigns distinct colors to any two different vertices incident with the same face.

Borodin–Broersma–Glebov–van den Heuvel conjecture. Every plane graph $G$ with $\Delta^*(G)$ and $k^*$ sufficiently large has a cyclic coloring with

$$\Delta^*(G)+k^*$$

colors.

This conjecture refines the general cyclic-coloring problem by incorporating the overlap parameter $k^*$. The source presents it as an open question posed by Borodin, Broersma, Glebov, and van den Heuvel; no resolution is supplied here.