A counterexample to a conjecture on facial uniquemaximal colorings
Abstract
A facial uniquemaximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. 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. Thus we conclude that facial uniquemaximum chromatic number of the sphere is five.
 Publication:

arXiv eprints
 Pub Date:
 September 2017
 arXiv:
 arXiv:1709.04958
 Bibcode:
 2017arXiv170904958L
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 4 pages, 2 figures