Two adjacent edges of a plane graph are facially adjacent if they are consecutive in a cyclic order around their common end vertex. A coloring is proper if adjacent vertices (facially adjacent edges) are colored differently.
Unique maximum and minimum (double maximum) coloring of a plane graph $G$ is a proper UM coloring of $G$ in which for each face there is by the lowest (second highest) color of that face colored exactly one element.
In this talk, we present upper bounds on the chromatic numbers for these four types of UM coloring.
For unique maximum and minimum coloring we proved that each plane graph is 7-vertex-colorable and 7-edge-colorable and similarly for unique double maximum coloring that each plane graph is 10-vertex-colorable and 7-edge-colorable.