The regular number of a graph

TL;DR

This paper investigates the regular number of graphs, which is the minimum partition of edges into regular subgraphs, providing exact values for certain graph families and establishing bounds that are sometimes tight.

Contribution

It determines the regular number for specific graph families and confirms the tightness of some previously known bounds.

Findings

Exact regular number for some graph families

General bounds on the regular number

Certain bounds are shown to be tight

Abstract

Let $G$ be a simple undirected graph. The regular number of $G$ is defined to be the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is regular. In this work, we obtain the regular number of some families of graphs and discuss some general bounds on this parameter. Also, some of the lower or upper bounds proved in \cite{Kulli:Janakiram:Iyer:2001} are shown here to hold with equality.