A note on internal partitions: the $5$-regular case and beyond

TL;DR

This paper investigates internal partitions in 5-regular graphs, showing certain subgraph properties and characterizing Abelian Cayley graphs that lack such partitions, extending understanding beyond known cases.

Contribution

It advances the study of internal partitions by analyzing 5-regular graphs, identifying subgraph intersections, and classifying specific Cayley graphs without internal partitions.

Findings

Existence of subgraphs with minimum degree at least 3 and small intersection

Characterization of 5-regular Abelian Cayley graphs without internal partitions

Extension of known results from 3, 4, 6-regular graphs to the 5-regular case

Abstract

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the $5$-regular case and show that among the subgraphs of minimum degree at least $3$, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all $5$-regular Abelian Cayley graphs which do not have an internal partition.