3-facial colouring of plane graphs

Frédéric Havet 1 Jean-Sébastien Sereni 1 Riste Skrekovski 2
1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
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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Frédéric Havet, Jean-Sébastien Sereni, Riste Skrekovski. 3-facial colouring of plane graphs. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2008, 22 (1), pp.231--247. ⟨10.1137/060664124⟩. ⟨inria-00083533v4⟩

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