The Hamilton-Waterloo Problem HWP(v;m,n;α,β) asks for a
2-factorization of the complete graph Kv or Kv−I, the complete graph with
the edges of a 1-factor removed, into αCm-factors and βCn-factors, where 3≤m<n. In the case that m and n are both
even, the problem has been solved except possibly when 1∈{α,β}
or when α and β are both odd, in which case necessarily v≡2(mod4). In this paper, we develop a new construction that creates
factorizations with larger cycles from existing factorizations under certain
conditions. This construction enables us to show that there is a solution to
HWP(v;2m,2n;α,β) for odd α and β whenever the obvious
necessary conditions hold, except possibly if β=1; β=3 and
gcd(m,n)=1; α=1; or v=2mn/gcd(m,n). This result almost completely
settles the existence problem for even cycles, other than the possible
exceptions noted above