Geodesic
Short cycles in random regular graphs
The electronic journal of combinatorics, Tome 11 (2004) no. 1
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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.
DOI : 10.37236/1819
Classification : 05C80, 05C38 
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     author = {Brendan D. McKay and Nicholas C. Wormald and Beata Wysocka},
     title = {Short cycles in random regular graphs},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {1},
     doi = {10.37236/1819},
     zbl = {1063.05122},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1819/}
}
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Brendan D. McKay; Nicholas C. Wormald; Beata Wysocka. Short cycles in random regular graphs. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1819

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