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Random Regular Graphs: Asymptotic Distributions and Contiguity

Published online by Cambridge University Press:  12 September 2008

Svante Janson
Svante Janson
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Swedensvante.janson@math.uu.se
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Abstract

The asymptotic distribution of the number of Hamilton cycles in a random regular graph is determined. The limit distribution is of an unusual type; it is the distribution of a variable whose logarithm can be written as an infinite linear combination of independent Poisson variables, and thus the logarithm has an infinitely divisible distribution with a certain discrete Lévy measure. Similar results are found for some related problems. These limit results imply that some different models of random regular graphs are contiguous, which means that they are qualitatively asymptotically equivalent. For example, if r > 3, then the usual (uniformly distributed) random r-regular graph is contiguous to the one constructed by taking the union of r perfect matchings on the same vertex set (assumed to be of even cardinality), conditioned on there being no multiple edges. Some consequences of contiguity for asymptotic distributions are discussed.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 4 , December 1995 , pp. 369 - 405
Copyright
Copyright © Cambridge University Press 1995

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