TITLE: expected length of a largest cycle in regular graph
QUESTION [5 upvotes]: Consider a simple random regular graph $G_d(n)$ with $d=2$ (that is, I select a $2-$regular graph from the set of all $2-$regular graphs on $n$ vertices, uniformly at random).
It is clear that this graph consists of (not-intersecting) cycles only; and asymptotically a.s., it does not contain a cycle of length $n$.
My question is: Can we characterize the length of its largest cycle?

REPLY [3 votes]: I'm not going to do this calculation, but here is how it can be done. Any interval which contains the length of the longest cycle with probability tending to 1 will also contain the length of the longest cycle of a random pairing (configuration) with probability tending to 1. And vice-versa. With a random pairing you can easily calculate the expectation of the number of sequences of cycles with given lengths. From this the distribution of the number of cycles longer than a given length will follow. I'm not sure how messy the details will be, but it's easier than usual calculations in the pairing model because cycles cannot partially overlap.