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On facial unique-maximum (edge-)coloring
A facial unique-maximum coloring of a plane graph is a vertex coloring where
on each face the maximal color appears exactly once on the vertices of
. If the coloring is required to be proper, then the upper bound for
the minimal number of colors required for such a coloring is set to .
Fabrici and G\"oring [Fabrici and Goring 2016] even conjectured that colors
always suffice. Confirming the conjecture would hence give a considerable
strengthening of the Four Color Theorem. In this paper, we prove that the
conjecture holds for subcubic plane graphs, outerplane graphs and plane
quadrangulations. Additionally, we consider the facial edge-coloring analogue
of the aforementioned coloring and prove that every -connected plane graph
admits such a coloring with at most colors.Comment: 5 figure
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