Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V , let c(v) denote the sum of the weight of v ∈ V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) ≠ c(v) for any two adjacent vertices u and v of G. We show that for every 2-connected plane graph there is such a proper vertex coloring with weights in {1, 2, 3}. In a special case, the value 3 is improved to 2.
@article{bwmeta1.element.doi-10_7151_dmgt_1771,
author = {Igor Fabricia and Stanislav Jendrol' and Roman Sot\'ak},
title = {A Note On Vertex Colorings Of Plane Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {849-855},
zbl = {1303.05037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1771}
}
Igor Fabricia; Stanislav Jendrol’; Roman Soták. A Note On Vertex Colorings Of Plane Graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 849-855. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1771/
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