Facial colorings using Hall's Theorem

Authors:
Frédéric Havet;Daniel Král';Jean-Sébastien Sereni;Riste Škrekovski
Affiliations:
MASCOTTE, I3S (CNRS-UNSA)-INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France;Institute for Theoretical Computer Science (ITI), Faculty of Mathematics and Physics, Charles University, Malostranské námstí 25, 11800 Prague, Czech Republic;CNRS (LIAFA, Université Denis Diderot), Paris, France and Department of Applied Mathematics (KAM), Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Mathematics, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia
Venue:
European Journal of Combinatorics
Year:
2010

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Abstract

A vertex coloring of a plane graph is @?-facial if every two distinct vertices joined by a facial walk of length at most @? receive distinct colors. It has been conjectured that every plane graph has an @?-facial coloring with at most 3@?+1 colors. We improve the currently best known bound and show that every plane graph has an @?-facial coloring with at most @?7@?/2@?+6 colors. Our proof uses the standard discharging technique, however, in the reduction part we have successfully applied Hall's Theorem, which seems to be quite an unusual approach in this area.