Beyond the MaxCut problem in $H$-free graphs

TL;DR

This paper strengthens existing bounds on maximum cuts in $H$-free graphs and graphs with no large cliques, providing new insights into graph partitioning and structural properties.

Contribution

It extends Zhang's methods to prove larger cuts in graphs without large cliques and characterizes graphs close to disjoint unions of cliques based on MaxCut and eigenvalues.

Findings

Graphs without large cliques have cuts significantly larger than half the edges.

Graphs with small MaxCut are structurally close to disjoint unions of cliques.

New bounds relate spectral properties to graph structure.

Abstract

In a recent breakthrough, Zhang proves that if $G$ is an $H$-free graph with $m$ edges, then $G$ has a cut of size at least $m/2+c_Hm^{0.5001}$, making a significant step towards a well known conjecture of Alon, Bollob\'as, Krivelevich and Sudakov. We show that the methods of Zhang can be further boosted, and prove the following strengthening. If $G$ is a graph with $m$ edges and no clique of size $m^{1/2-\delta}$, then $G$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(\delta)>0$. In addition, we sharpen another result of Zhang by proving that if $G$ is an $n$-vertex $m$-edge graph with MaxCut of size at most $m/2+n^{1+\varepsilon}$ (or its smallest eigenvalue $\lambda_n$ satisfies $|\lambda_n|\leq n^{\varepsilon}$), then $G$ is $n^{-\varepsilon}$-close to the disjoint union of cliques for some absolute constant $\varepsilon>0$.