3-facial colouring of plane graphs
<|reference_start|>3-facial colouring of plane graphs: 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.<|reference_end|>
@article{havet20063-facial,
    title={3-facial colouring of plane graphs},
    author={F'ed'eric Havet (INRIA Sophia Antipolis), Jean-S'ebastien Sereni
  (INRIA Sophia Antipolis), Riste Skrekovski},
    journal={arXiv preprint arXiv:cs/0607008},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0607008},
    primaryClass={cs.DM}
}