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

• Published in: Discrete mathematics Vol. 341; no. 6; pp. 1636 - 1644
• Main Authors: Burgess, A.C., Danziger, P., Traetta, T.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.06.2018
• Subjects: 2-factorizations; Cycle systems; Generalized Oberwolfach Problem; Hamilton–Waterloo problem; Resolvable cycle decompositions; Resolvable cycle decompositions; Hamilton–Waterloo problem; Generalized Oberwolfach Problem; 2-factorizations; Cycle systems
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2018.02.020

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.