Given a weighting of all elements of a 2-connected plane graph G = (V,E, F), let f(α) denote the sum of the weights of the edges and vertices incident with the face _ and also the weight of _. Such an entire weighting is a proper face colouring provided that f(α) ≠ f(β) for every two faces α and _ sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3
@article{bwmeta1.element.doi-10_7151_dmgt_1738,
author = {Stanislav Jendrol and Peter \v Sugerek},
title = {A note on face coloring entire weightings of plane graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {421-426},
zbl = {1290.05065},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1738}
}
Stanislav Jendrol; Peter Šugerek. A note on face coloring entire weightings of plane graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 421-426. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1738/
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