Max-Bisections of graphs without perfect matching

TL;DR

This paper demonstrates that in certain graphs without specific even cycles, large bisections can be guaranteed using minimum degree conditions instead of perfect matching conditions, confirming a recent conjecture.

Contribution

It replaces the perfect matching condition with a minimum degree condition for large bisections in graphs without certain even cycles, confirming a conjecture by Lin and Zeng.

Findings

Graphs with minimum degree at least 2 have large bisections proportional to the number of edges.

Confirmed a conjecture relating forbidden cycles and bisection size.

Provides bounds on bisection size based on forbidden cycles and graph parameters.

Abstract

A bisection of a graph is a bipartition of its vertex set such that the two resulting parts differ in size by at most 1, and its size is the number of edges that connect vertices in the two parts. The perfect matching condition and forbidden even cycles subgraphs are essential in finding large bisections of graphs. In this paper, we show that the perfect matching condition can be replaced by the minimum degree condition. Let $C_{\ell}$ be a cycle of length $\ell$ for $\ell\ge 3$, and let $G$ be a $\{C_4, C_6\}$-free graph with $m$ edges and minimum degree at least 2. We prove that $G$ has a bisection of size at least $m/2+\Omega\left(\sum_{v\in V(G)}\sqrt{d(v)}\right)$. As a corollary, if $G$ is also $C_{2k}$-free for $k\ge3$, then $G$ has a bisection of size at least $m / 2+\Omega\left(m^{(2 k+1) /(2 k+2)}\right)$, thereby confirming a conjecture proposed by Lin and Zeng [J. Comb. Theory A, 180 (2021), 105404].