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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, [email protected]
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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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References

[1]Barbour, A. D., Janson, S., Karoński, M. and A., Ruciński (1990) Small cliques in random graphs. Random Struct. Alg. 1 403–434.CrossRefGoogle Scholar
[2]Bender, E. A. and Canfield, E. R. (1978) The asymptotic number of labeled graphs with given degree sequences. J. Combin. Th. A 24 296–307.Google Scholar
[3]Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
[4]Bollobás, B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combinatorics 1 311–316.CrossRefGoogle Scholar
[5]Bollobás, B. (1985) Random Graphs Academic Press.Google Scholar
[6]Bollobás, B. and McKay, B. D. (1986) The number of matchings in random regular graphs and bipartite graphs. J. Combin. Th. B 41 80–91.Google Scholar
[7]Cooper, C., Frieze, A., and Molloy, M. (1994) Hamilton cycles in random regular digraphs. Combinatorics, Probability and Computing 3 39–50.Google Scholar
[8]Cooper, C., Frieze, A., Molloy, M. and Reed, B. (to appear) Perfect matchings in random r-regular, s-uniform hypergraphs. Combinatorics, Probability and Computing.Google Scholar
[9]Frieze, A.M., Jerrum, M.R., Molloy, M., Robinson, R. and Wormald, N. (to appear) Generating and counting Hamilton cycles in random regular graphs. Journal of Algorithms.Google Scholar
[10]Frieze, A. and Suen, S. (1992) Counting Hamilton cycles in random directed graphs. Random Struct. Alg. 3 235–242CrossRefGoogle Scholar
[11]Janson, S. (1994) Orthogonal decompositions and functional limit theorems for random graph statistics. Memoirs Amer. Math. Soc. 534Google Scholar
[12]Janson, S. (1994) The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph. Combinatorics, Probability and Computing 3 97–126CrossRefGoogle Scholar
[13]Le Cam, L. (1960) Locally asymptotically normal families of distributions. Univ. of California Publ. in Statistics 3 37–98.Google Scholar
[14]Le Cam, L. (1969) Théorie asymptotique de la décision statistique. Les Presses de l'université de Montréal.Google Scholar
[15]Le Cam, L. (1986) Asymptotic methods in statistical decision theory. Springer-Verlag.CrossRefGoogle Scholar
[16]Loève, M. (1977) Probability theory 4th ed.Springer-Verlag.Google Scholar
[17]Molloy, M., Robalewska-Szarłat, H., Robinson, R.W. and Wormald, N.C. (to appear) 1-factorisations of random regular graphs.Google Scholar
[18]Robalewska-Szarłat, H. (to appear) 2-factors in random regular graphs.Google Scholar
[19]Robinson, R.W. and Wormald, N.C. (1984) Existence of long cycles in random cubic graphs. Enumeration and Design, Academic Press, pp. 251–270Google Scholar
[20]Robinson, R.W. and Wormald, N.C. (1992) Almost all cubic graphs are hamiltonian. Random Struct. Alg. 3 117–125CrossRefGoogle Scholar
[21]Robinson, R.W. and Wormald, N. C. (1994) Almost all regular graphs are hamiltonian. Random Struct. Alg. 5 363–374CrossRefGoogle Scholar
[22]Roussas, G. (1972) Contiguity of probability measures: some applications in statistics, Cambridge University Press.CrossRefGoogle Scholar
[23]Skorokhod, A. V. (1956) Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen. 1 289–319 (in Russian) (English transl., Theor. Probab. Appl. 1 (1956) 261–290.)Google Scholar
[24]Wormald, N.C. (1981) The asymptotic distribution of short cycles in random regular graphs. J. Combin. Th. B31 168–182CrossRefGoogle Scholar