Max-Bisections of graphs without even cycles

TL;DR

This paper extends the Alon-Krivelevich-Sudakov Theorem to show that graphs without even cycles and with minimum degree at least k contain balanced bipartite subgraphs with significantly many edges, improving understanding of graph bisections.

Contribution

It provides a new bisection variant of the theorem, answering a specific open problem and enhancing recent related results, based on bounds for graph bisections with sparse neighborhoods.

Findings

Graphs without even cycles have large balanced bipartite subgraphs.

Minimum degree condition ensures a high edge count in bipartite subgraphs.

The approach uses bounds related to degree sequences and sparse neighborhoods.

Abstract

For an integer $k\ge 2$, let $G$ be a graph with $m$ edges and without cycles of length $2k$. The pivotal Alon-Krivelevich-Sudakov Theorem on Max-Cuts states that $G$ has a bipartite subgraph with at least $m/2+\Omega(m^{(2k+1)/(2k+2)})$ edges. In this paper, we present a bisection variant of it by showing that if $G$ has minimum degree at least $k$, then $G$ has a balanced bipartite subgraph with at least $m/2+\Omega(m^{(2k+1)/(2k+2)})$ edges. It not only answers a problem of Fan, Hou and Yu in full generality but also enhances a recent result given by Hou, Wu and Zhong. Our approach hinges on a key bound for bisections of graphs with sparse neighborhoods concerning the degree sequence. The result is inspired by the celebrated approximation algorithm of Goemans and Williamson and appears to be worthy of future exploration.