Balanced 1-Factorisations of 3- and 4-Regular Circulant Graphs

TL;DR

This paper studies balanced 1-factorisations in 3- and 4-regular circulant graphs, focusing on equal occurrence of 2-regular unions across pairs of 1-factors, contributing new insights into their structure.

Contribution

It introduces the concept of balanced 1-factorisations and provides initial results on their existence in 3- and 4-regular circulant graphs.

Findings

Identifies conditions for balanced 1-factorisations in these graphs

Establishes existence results for certain classes of circulant graphs

Provides a framework for further exploration of balanced factorizations

Abstract

We investigate 1-factorisations in which the 2-regular graphs that occur as the union of a pair of 1-factors appear an equal number of times across the unions of all pairs of 1-factors in the 1-factorisation. We call such 1-factorisations balanced 1-factorisations (B1Fs) and we present some results on B1Fs of 3- and 4-regular circulant graphs.