On regular hypergraphs of high girth
Abstract
We give lower bounds on the maximum possible girth of an -uniform, -regular hypergraph with at most vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute constant factor (viz., by a factor of between and ). We also define a random -uniform `Cayley' hypergraph on which has girth with high probability, in contrast to random regular -uniform hypergraphs, which have constant girth with positive probability.
Keywords
Cite
@article{arxiv.1302.5090,
title = {On regular hypergraphs of high girth},
author = {David Ellis and Nathan Linial},
journal= {arXiv preprint arXiv:1302.5090},
year = {2017}
}
Comments
A hole in the proof of Theorem 8 was recently pointed out by Eberhard [10]. In this version, we repair the hole, giving a slightly weaker bound. We use a very similar method to that of Eberhard in [10], where a slightly weaker variant of [13, Theorem 3] is proved; our original proof contained a similar hole to the original proof of [13, Theorem 3]