We consider a simple, undirected graph G. The ball of a subset Y of vertices in G is the set of vertices in G at distance at most one from a vertex in Y. Assuming that the balls of all subsets of at most two vertices in G are distinct, we prove that G admits a cycle with length at least 7.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1516,
author = {David Auger and Ir\`ene Charon and Olivier Hudry and Antoine Lobstein},
title = {On the existence of a cycle of length at least 7 in a (1,$\leq$ 2)-twin-free graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {591-609},
zbl = {1217.05116},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1516}
}
David Auger; Irène Charon; Olivier Hudry; Antoine Lobstein. On the existence of a cycle of length at least 7 in a (1,≤ 2)-twin-free graph. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 591-609. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1516/
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