Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d-1)2g-1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.
|Number of pages||12|
|Journal||Electronic Journal of Combinatorics|
|Issue number||1 R|
|Publication status||Published - 20 Sep 2004|