Suppose $ S_{n,N} $ be the set of $n$ elements with $N$ many cycles where $N$ is proportional to $n$. $U_{n,N}$ is an element picked randomly from this. It is known that the length of any cycle cannot be too large. What if we restrict the cycle lengths to be odd and then pick uniformly from this? Should it be also be that there are no cycles with huge length w.h.p?

I can prove this is the case if cycle lengths are all even. Is there any known reference for this?

Thanks