On the Hamilton–Waterloo Problem with cycle lengths of distinct parities

Abstract

Let Kv∗ denote the complete graph Kv if v is odd and Kv−I, the complete graph with the edges of a 1-factor removed, if v is even. Given non-negative integers v,M,N,α,β, the Hamilton–Waterloo problem asks for a 2-factorization of Kv∗ into αCM-factors and βCN-factors. Clearly, M,N≥3, M∣v, N∣v and α+β=⌊v−12⌋ are necessary conditions.Very little is known on the case where M and N have different parities. In this paper, we make some progress on this case by showing, among other things, that the above necessary conditions are sufficient whenever M|N, v>6N>36M, and β≥3.