We consider a vertex colouring of a connected plane graph G. A colour c is used k times by a face α of G if it appears k times along the facial walk of α. We prove that every connected plane graph with minimum face degree at least 3 has a vertex colouring with four colours such that every face uses some colour an odd number of times. We conjecture that such a colouring can be done using three colours. We prove that this conjecture is true for 2-connected cubic plane graphs. Next we consider other three kinds of colourings that require stronger restrictions.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1462,
author = {J\'ulius Czap and Stanislav Jendrol'},
title = {Colouring vertices of plane graphs under restrictions given by faces},
journal = {Discussiones Mathematicae Graph Theory},
volume = {29},
year = {2009},
pages = {521-543},
zbl = {1193.05065},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1462}
}
Július Czap; Stanislav Jendrol'. Colouring vertices of plane graphs under restrictions given by faces. Discussiones Mathematicae Graph Theory, Tome 29 (2009) pp. 521-543. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1462/
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