A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4, ≤ 13) or (5, ≤ 9) or (6, ≤ 8) or (7,7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each of the considered edge types there are 1-planar graphs whose set of types of edges contains just the selected edge type).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1625,
author = {D\'avid Hud\'ak and Peter \v Sugerek},
title = {Light edges in 1-planar graphs with prescribed minimum degree},
journal = {Discussiones Mathematicae Graph Theory},
volume = {32},
year = {2012},
pages = {545-556},
zbl = {1262.05032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1625}
}
Dávid Hudák; Peter Šugerek. Light edges in 1-planar graphs with prescribed minimum degree. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 545-556. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1625/
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