Asymptotically Almost Every $2r$-regular Graph has an Internal Partition

TL;DR

This paper proves that almost all large 2r-regular graphs can be partitioned so that each vertex has at least half of its neighbors on its side, revealing a widespread structural property.

Contribution

It establishes that asymptotically almost every 2r-regular graph admits an internal partition, a significant step in understanding graph partition properties.

Findings

Almost all large 2r-regular graphs have an internal partition.

The result holds asymptotically as the number of vertices grows.

Provides insight into the structure of regular graphs.

Abstract

An internal partition of a graph is a partitioning of the vertex set into two parts such that for every vertex, at least half of its neighbors are on its side. We prove that for every positive integer $r$, asymptotically almost every $2r$-regular graph has an internal partition.