For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.
@article{bwmeta1.element.doi-10_7151_dmgt_1657,
author = {Mirka Miller and Joe Ryan and Zden\v ek Ryj\'a\v cek},
title = {Distance-Locally Disconnected Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {203-215},
zbl = {1293.05168},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1657}
}
Mirka Miller; Joe Ryan; Zdeněk Ryjáček. Distance-Locally Disconnected Graphs. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 203-215. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1657/
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