Hypergraph cuts above the average

TL;DR

This paper extends classical graph cut results to hypergraphs, showing that large cuts exceeding the expected size are common, with bounds depending on hypergraph parameters, revealing new behavior for different uniformities and partitions.

Contribution

It establishes new bounds for the size of maximum cuts in hypergraphs, generalizing Edwards' result and identifying cases with significantly larger cuts than expected.

Findings

For k=3, r=2, the bound is tight and achieved by Steiner triple systems.

For other cases (k≥4 or r≥3), the excess over expected cut size is Ω(m^{5/9}).

Large cuts exceeding the expected size are common in hypergraphs, with bounds depending on parameters.

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 $m/2 + \Omega(\sqrt{m})$, 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 $\Omega(\sqrt m)$ 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 $k \geq 4$ or $r \geq 3$), we show that every m-edge k-uniform hypergraph has an r-cut whose size is $\Omega(m^{5/9})$ 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.