Max-Bisections of graphs without even cycles

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.