Bisections of graphs

TL;DR

This paper extends classical bounds on maximum cuts to bisections, providing new results on the existence of balanced bipartitions with many crossing edges and few internal edges, addressing conjectures by Bollobás and Scott.

Contribution

It generalizes Edwards's bound to bisections, proves a conjecture on minimum degree graphs, and introduces tools for analyzing balanced bipartitions.

Findings

Every graph with no isolated vertices and degree ≤ n/3+1 has a bisection of size ≥ m/2 + n/6.

Graphs with large minimum degree have bisections with few internal edges.

Confirmed a conjecture that graphs with minimum degree ≥ 2 have balanced bisections with at most (1/3+o(1))m internal edges.

Abstract

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollob\'as and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A simple corollary of our result implies that every graph on $n$ vertices and $m$ edges with no isolated vertices, and maximum degree at most $n/3 + 1$, admits a bisection of size at least $m/2 + n/6$. Then using the tools that we developed to extend Edwards's bound, we prove a judicious bisection result which states that graphs with large minimum degree have a bisection in which both parts span relatively few edges. A special case of this general theorem answers a conjecture of Bollob\'as and Scott, and shows that every graph on $n$ vertices and $m$ edges of minimum degree at least 2 admits a bisection in which the number of edges in each part is at most $(1/3+o(1))m$. We also present several other results on bisections of graphs.