Hypergraph cuts above the average

Conlon D, Fox J, Kwan MA, Sudakov B. 2019. Hypergraph cuts above the average. Israel Journal of Mathematics. 233(1), 67–111.


Journal Article | Published | English

Scopus indexed
Author
Conlon, David; Fox, Jacob; Kwan, Matthew AlanIST Austria; Sudakov, Benny
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(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.
Publishing Year
Date Published
2019-08-01
Journal Title
Israel Journal of Mathematics
Volume
233
Issue
1
Page
67-111
ISSN
eISSN
IST-REx-ID

Cite this

Conlon D, Fox J, Kwan MA, Sudakov B. Hypergraph cuts above the average. Israel Journal of Mathematics. 2019;233(1):67-111. doi:10.1007/s11856-019-1897-z
Conlon, D., Fox, J., Kwan, M. A., & Sudakov, B. (2019). Hypergraph cuts above the average. Israel Journal of Mathematics. Springer. https://doi.org/10.1007/s11856-019-1897-z
Conlon, David, Jacob Fox, Matthew Alan Kwan, and Benny Sudakov. “Hypergraph Cuts above the Average.” Israel Journal of Mathematics. Springer, 2019. https://doi.org/10.1007/s11856-019-1897-z.
D. Conlon, J. Fox, M. A. Kwan, and B. Sudakov, “Hypergraph cuts above the average,” Israel Journal of Mathematics, vol. 233, no. 1. Springer, pp. 67–111, 2019.
Conlon D, Fox J, Kwan MA, Sudakov B. 2019. Hypergraph cuts above the average. Israel Journal of Mathematics. 233(1), 67–111.
Conlon, David, et al. “Hypergraph Cuts above the Average.” Israel Journal of Mathematics, vol. 233, no. 1, Springer, 2019, pp. 67–111, doi:10.1007/s11856-019-1897-z.
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