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Hypergraph cuts above the average

Conlon, David and Fox, Jacob and Kwan, Matthew and Sudakov, Benny (2019) Hypergraph cuts above the average. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20190819-170914451

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Abstract

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2 + Ω(√m), and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is Ω(√m) larger than the expected size of a random r-cut. Moreover, in the case where k = 3 and r = 2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if k ≥ 4 or r ≥ 3), we show that every m-edge k-uniform hypergraph has an r-cut whose size is Ω(m^(5/9)) larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1803.08462arXivDiscussion Paper
ORCID:
AuthorORCID
Conlon, David0000-0001-5899-1829
Additional Information:Conlon research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. Fox research supported by a Packard Fellowship and by NSF Career Award DMS-1352121. Sudakov research supported in part by SNSF grant 200021-175573.
Funders:
Funding AgencyGrant Number
Royal SocietyUNSPECIFIED
European Research Council (ERC)676632
David and Lucile Packard FoundationUNSPECIFIED
NSFDMS-1352121
Swiss National Science Foundation (SNSF)200021-175573
Record Number:CaltechAUTHORS:20190819-170914451
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190819-170914451
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98027
Collection:CaltechAUTHORS
Deposited By: Melissa Ray
Deposited On:20 Aug 2019 19:49
Last Modified:03 Oct 2019 21:37

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