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{\bf Brendan D. McKay, Nicholas C. Wormald and Beata Wysocka}
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{\bf Short Cycles in Random Regular Graphs}
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Consider random regular graphs of order $n$ and degree $d=d(n)\ge 3$.
Let $g=g(n)\ge 3$ 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.
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