Internal Partitions of Regular Graphs

TL;DR

This paper proves that all 6-regular graphs with sufficiently many vertices have an internal partition, advances understanding of such partitions in regular graphs, and explores related bounds and open problems.

Contribution

It establishes the existence of internal partitions for 6-regular graphs with large vertex counts and introduces new bounds and graph families related to internal partitions.

Findings

Proved all 6-regular graphs with enough vertices have internal partitions.

Provided new lower bounds on the function N(d) for regular graphs.

Identified new graph families without internal partitions.

Abstract

An internal partition of an $n$-vertex graph $G=(V,E)$ is a partition of $V$ such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every $d$-regular graph with $n>N(d)$ vertices has an internal partition. Here we prove this for $d=6$. The case $d=n-4$ is of particular interest and leads to interesting new open problems on cubic graphs. We also provide new lower bounds on $N(d)$ and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well.