Bisections of graphs under degree constraints
Jie Ma, Hehui Wu
TL;DR
The paper addresses the problem of finding bisections of general graphs under degree constraints by developing a unified two-stage framework that combines a random partition with a deterministic refinement. It proves two main universal results: every graph has a bisection in which each vertex $v$ has at least $\left(\frac{1}{4}-o(1)\right)d_G(v)$ neighbors in its own part and another where each vertex has at least $\left(\frac{1}{4}-o(1)\right)d_G(v)$ neighbors in the opposite part, which are asymptotically tight up to a factor of $1/2$. The authors then derive a function $f(k)=O(k)$ such that graphs with minimum degree at least $f(k)$ admit a bisection with every vertex having at least $k$ neighbors in its own part and a complementary bisection with at least $k$ neighbors in the opposite part, alongside stronger results ensuring most vertices meet these external/internal targets with minor relaxations. These findings extend Erdős–Thomassen–Kühn–Osthus results, provide a systematic density-based approach to partitioning under degree constraints, and open avenues for tightening constants and generalizing to multi-partitions and connectivity.
Abstract
In this paper, we investigate the problem of finding {\it bisections} (i.e., balanced bipartitions) in graphs. We prove the following two results for {\it all} graphs $G$: (1). $G$ has a bisection where each vertex $v$ has at least $(1/4 - o(1))d_G(v)$ neighbors in its own part; (2). $G$ also has a bisection where each vertex $v$ has at least $(1/4 - o(1))d_G(v)$ neighbors in the opposite part. These results are asymptotically optimal up to a factor of $1/2$, aligning with what is expected from random constructions, and provide the first systematic understanding of bisections in general graphs under degree constraints. As a consequence, we establish for the first time the existence of a function $f(k)$ such that for any $k\geq 1$, every graph with minimum degree at least $f(k)$ admits a bisection where every vertex has at least $k$ neighbors in its own part, as well as a bisection where every vertex has at least $k$ neighbors in the opposite part. Using a more general setting, we further show that for any $\varepsilon > 0$, there exist $c_\varepsilon, c'_\varepsilon > 0$ such that any graph $G$ with minimum degree at least $c_\varepsilon k$ (respectively, $c'_\varepsilon k$) admits a bisection satisfying: every vertex has at least $k$ neighbors in its own part (respectively, in the opposite part), and at least $(1 - \varepsilon)|V(G)|$ vertices have at least $k$ neighbors in the opposite part (respectively, in their own part). These results extend and strengthen classical graph partitioning theorems of Erdős, Thomassen, and Kühn-Osthus, while additionally satisfying the bisection requirement.