Doyen-Wilson results for odd length cycle systems

TL;DR

This paper fully characterizes when odd-length cycle systems can be embedded into larger systems, resolving most cases and providing necessary and sufficient conditions for cycle decompositions with holes.

Contribution

It extends Doyen-Wilson results by providing complete solutions for embedding odd cycle systems and establishing conditions for cycle decompositions with holes.

Findings

Complete solutions for embedding odd cycle systems for large or prime power m

Necessary and sufficient conditions for cycle decompositions with holes

Identification of exceptions when v-u is small compared to m

Abstract

For each odd $m \geq 3$ we completely solve the problem of when an $m$-cycle system of order $u$ can be embedded in an $m$-cycle system of order $v$, barring a finite number of possible exceptions. In cases where $u$ is large compared to $m$, where $m$ is a prime power, or where $m \leq 15$, the problem is completely resolved. In other cases, the only possible exceptions occur when $v-u$ is small compared to $m$. This result is proved as a consequence of a more general result which gives necessary and sufficient conditions for the existence of an $m$-cycle decomposition of a complete graph of order $v$ with a hole of size $u$ in the case where $u \geq m-2$ and $v-u \geq m+1$ both hold.