A counterexample to a conjecture on facial unique-maximal colorings
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A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face α the maximal color appears exactly once on the vertices of α. Fabrici and Göring  proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color theorem. Wendland  later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. s we conclude that facial unique-maximum chromatic number of the sphere is five.
This is a manuscript of an article published as Lidický, Bernard, Kacy Messerschmidt, and Riste Škrekovski. "A counterexample to a conjecture on facial unique-maximal colorings." Discrete Applied Mathematics 237 (2018): 123-125. doi: 10.1016/j.dam.2017.11.037. Posted with permission.