A counterexample to a conjecture on facial unique-maximal colorings

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2018-03-11
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Lidicky, Bernard
Messerschmidt, Kacy
Lidicky, Bernard
Skrekovski, Riste
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Mathematics
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Mathematics
Abstract

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 [4] 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 [6] 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.

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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.

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