It is conjectured that for every pair (l,m) of odd integers greater than 2 with m=1 (mod l), there exists a cyclic two-factorization of Klm having exactly (m-1)/2 factors of type lm and all the others of type ml. The authors prove the conjecture in the affirmative when l = 1(mod4) and m≥l2-l+1.
Merola, F., Traetta, T. (2016). Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles. DISCRETE MATHEMATICS, 339(9), 2267-2283 [10.1016/j.disc.2016.03.026].