A graph partition problem

TL;DR

This paper investigates the conditions under which the edge set of an m-fold complete graph can be partitioned into copies of a given graph G, introducing the concepts of partition modulus and partition index.

Contribution

It establishes the existence of a partition modulus for any graph G and explores the properties of the partition index, including cases where it differs from 1, along with computations for specific graphs.

Findings

Existence of a finite partition modulus m_0 for graph G.

Most graphs have a partition index of 1, but some do not.

Connections between graph partitions and combinatorial design theory.

Abstract

Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we give two infinite families of graphs for which this is not true. We also compute $M(G)$ for various graphs, and outline some connections between our problem and the existence of designs of various types.