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Avoiding long Berge cycles
[article]
<span title="2018-05-14">2018</span>
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arXiv
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<span class="release-stage" >pre-print</span>
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<span title="2018-05-10">2018</span>
<span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.04195v1">arXiv:1805.04195v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cwz3gksvejbfvj6v5jakbjns3y">fatcat:cwz3gksvejbfvj6v5jakbjns3y</a> </span>
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Let n≥ k≥ r+3 and H be an n-vertex r-uniform hypergraph. We show that if | H|> n-1/k-2k-1r then H contains a Berge cycle of length at least k. This bound is tight when k-2 divides n-1. We also show that the bound is attained only for connected r-uniform hypergraphs in which every block is the complete hypergraph K^(r)_k-1. We conjecture that our bound also holds in the case k=r+2, but the case of short cycles, k≤ r+1, is different.
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