Doyen–Wilson Results for Odd Length Cycle Systems

For each odd m ≥ 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 ≤ 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 that 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 ≥ m − 2 and v − u ≥ m + 1 both hold.