# Edges on cycles in 3 regular graphs

Suppose I have a random 3 regular graph - there are many results about the expected number of cycles of given length in such a graph, and also about things like the probability that any two cycles of given length will intersect (the best resource I know of is a paper of McKay, Wormold and Wysocka, plus of course Bollobas). My question is the following - if I randomly choose an edge $e$, what is the expected number of cycles of some given length containing that edge? I guess the best way to formulate this might be to consider the set consisting of the number of (simple) cycles each edge is on, and ask is it known what the distribution of those numbers is?

Thanks

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The expected number of $k$-cycles can be calculated using the pairing (configuration) model and one of the references in the paper you mention will explain it. It has been calculated lots of times but I can't recall any of the places it is explicitly stated. The expected number of $k$-cycles using a random edge is $k/(3n/2)$ times the expected number of $k$-cycles altogether, by symmetry. The same approach will handle subgraphs other than cycles, for example a subgraph formed by two overlapping cycles. If you want the distribution (not just the expectation) of the number of $k$-cycles using an edge it gets trickier. You can get it (a Poisson distribution) for bounded $k$ by the pairing model. For slightly larger $k$, the switching method used in the paper you mention will also give a Poisson distribution. For quite a bit larger $k$ the methods of Garmo should work and the distribution will be more complicated. For very large cycles, it could be difficult. Sorry this answer is vague; I don't know where anyone has set this all out in a form you can just reference.