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Hypergraph cuts above the average

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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 \(m{\rm/}2 + {\rm\Omega)}(\sqrt m)\) 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 \({\rm\Omega}(\sqrt m)\) 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(m5/9) 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.

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Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    David Conlon

  2. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Jacob Fox & Matthew Kwan

  3. Department of Mathematics, ETH Zürich-Zentrum, CH-8092, Zürich, Switzerland

    Benny Sudakov

Authors
  1. David Conlon
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  2. Jacob Fox
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  3. Matthew Kwan
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  4. Benny Sudakov
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Corresponding author

Correspondence to Benny Sudakov.

Additional information

Research supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632.

Research supported by a Packard Fellowship and by NSF Career Award DMS-1352121.

This research was done while the author was working at ETH Zurich.

Research supported in part by SNSF grant 200021-175573.

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Conlon, D., Fox, J., Kwan, M. et al. Hypergraph cuts above the average. Isr. J. Math. 233, 67–111 (2019). https://doi.org/10.1007/s11856-019-1897-z

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  • DOI: https://doi.org/10.1007/s11856-019-1897-z

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