Large Cuts in Hypergraphs via Energy

TL;DR

This paper improves bounds on large r-partite subhypergraphs in hypergraphs, using advanced probabilistic, combinatorial, and algebraic techniques, and relates graph energy to maximum cut size.

Contribution

It establishes new lower bounds for r-partite subhypergraphs in hypergraphs, extending prior results and introducing novel energy-based methods.

Findings

For r ≥ 3, existence of r-partite subhypergraph with at least (r!/r^r)m + m^{3/5-o(1)} edges.

Improved bounds for linear hypergraphs to (r!/r^r)m + m^{3/4-o(1)} edges.

Every multigraph has a cut of size at least m/2 + Ω(energy/ log m).

Abstract

A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is $r=2$, the resulting bound $m/2$ can be improved to $m/2+\Omega(m^{1/2})$, and this is sharp. We prove that if $r\geq 3$, then there is an $r$-partite subhypergraph with at least $\frac{r!}{r^r} m+m^{3/5-o(1)}$ edges. Moreover, if the hypergraph is linear, this can be improved to $\frac{r!}{r^r} m+m^{3/4-o(1)},$ which is tight up to the $o(1)$ term. These improve results of Conlon, Fox, Kwan, and Sudakov. Our proof is based on a combination of probabilistic, combinatorial, and linear algebraic techniques, and semidefinite programming. A key part of our argument is relating the energy $\mathcal{E}(G)$ of a graph $G$ (i.e. the sum of absolute values of eigenvalues of the adjacency matrix) to its maximum cut. We prove that every $m$ edge multigraph $G$ has a cut of size at least $m/2+\Omega(\frac{\mathcal{E}(G)}{\log m})$, which might be of independent interest.