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3-facial colouring of plane graphs

Fédéric Havet·Jean-Sébastien Sereni·Riste Skrekovski
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Abstract

A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially 11-colourable. As a consequence, we derive that every 2-connected plane graph with maximum face-size at most 7 is cyclically 11-colourable. These two bounds are for one off from those that are proposed by the (3l+1)-Conjecture and the Cyclic Conjecture.

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Fédéric Havet, Jean-Sébastien Sereni, Riste Skrekovski. 2006-07-03. 3-facial colouring of plane graphs. https://arxiv.org/abs/cs/0607008

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