# Cyclic cycle systems of the complete multipartite graph

## TL;DR

This paper investigates the conditions under which complete multipartite graphs can be decomposed into cyclic cycles of a fixed length, providing a comprehensive characterization for certain divisibility cases.

## Contribution

It establishes necessary and sufficient conditions for cyclic -cycle decompositions of complete multipartite graphs when specific divisibility criteria are met.

## Key findings

- Derived necessary and sufficient conditions for existence
- Characterized decompositions for the case when 2 divides (m-1)n
- Extended understanding of cycle decompositions in multipartite graphs

## Abstract

In this paper, we study the existence problem for cyclic $\ell$-cycle decompositions of the graph $K_m[n]$, the complete multipartite graph with $m$ parts of size $n$, and give necessary and sufficient conditions for their existence in the case that $2\ell \mid (m-1)n$.