Facial colorings using Hall's Theorem

Abstract : A vertex coloring of a plane graph is ℓ-facial if every two distinct vertices joined by a facial walk of length at most ℓ receive distinct colors. It has been conjectured that every plane graph has an ℓ-facial coloring with at most 3ℓ+1 colors. We improve the currently best known bound and show that every plane graph has an ℓ-facial coloring with at most 7ℓ/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.
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Frédéric Havet, Daniel Král', Jean-Sébastien Sereni, Riste Skrekovski. Facial colorings using Hall's Theorem. European Journal of Combinatorics, Elsevier, 2010, 31 (3), pp.1001--1019. ⟨10.1016/j.ejc.2009.10.003⟩. ⟨hal-00487100⟩

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