Short cycles in random regular graphs

Brendan D. McKay; Nicholas C. Wormald; Beata Wysocka

Short cycles in random regular graphs

Brendan D. McKay, Nicholas C. Wormald, Beata Wysocka

Research output: Contribution to journal › Article › Research › peer-review

Abstract

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.

Original language	English
Pages (from-to)	1-12
Number of pages	12
Journal	Electronic Journal of Combinatorics
Volume	11
Issue number	1 R
Publication status	Published - 20 Sep 2004
Externally published	Yes

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abstract = "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.",

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N2 - 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.

AB - 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.

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