# Cycle packings of the complete multigraph

## TL;DR

This paper characterizes the conditions under which a complete multigraph can be packed with cycles of specified lengths, extending previous work on cycle decompositions to more general packings.

## Contribution

It provides a complete characterization for the existence of cycle packings with given cycle lengths in complete multigraphs, generalizing earlier decomposition results.

## Key findings

- Established necessary and sufficient conditions for cycle packings.
- Extended the theory from cycle decompositions to packings.
- Bridged the gap between decompositions and packings in multigraphs.

## Abstract

Bryant, Horsley, Maenhaut and Smith recently gave necessary and sufficient conditions for when the complete multigraph can be decomposed into cycles of specified lengths $m_1,m_2,\ldots,m_\tau$. In this paper we characterise exactly when there exists a packing of the complete multigraph with cycles of specified lengths $m_1,m_2,\ldots,m_\tau$. While cycle decompositions can give rise to packings by removing cycles from the decomposition, in general it is not known when there exists a packing of the complete multigraph with cycles of various specified lengths.