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On the Number of Perfect Matchings in Random Lifts


Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.

We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.

To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

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[1]Alon N., Benjamini I., Lubetzky E. and Sodin S. (2007) Non-backtracking random walks mix faster. Commun. Contemp. Math. 9 585603.
[2]Amit A. and Linial N. (2002) Random graph coverings I: General theory and graph connectivity. Combinatorica 22 118.
[3]Amit A. and Linial N. (2006) Random lifts of graphs: Edge expansion. Combin. Probab. Comput. 15 317332.
[4]Amit A., Linial N. and Matoušek J. (2002) Random lifts of graphs: Independence and chromatic number. Random Struct. Alg. 20 122.
[5]Angel O., Friedman J. and Hoory S. The non-backtracking spectrum of the universal cover of a graph. Preprint; available as arXiv:0712.0192v1 [math.CO].
[6]Burgin K., Chebolu P., Cooper C. and Frieze A. M. (2006) Hamilton cycles in random lifts of graphs. Europ. J. Combin. 27 12821293.
[7]Chebolu P. and Frieze A. M. (2008) Hamilton cycles in random lifts of complete directed graphs. SIAM J. Discrete Math. 22 520540.
[8]Greenhill C., Janson S., Kim J. H. and Wormald N. C. (2002) Permutation pseudographs and contiguity. Combin. Probab. Comput. 11 273298.
[9]Friedman J. (2008) A proof of Alon's second eigenvalue conjecture. Memoirs Amer. Math. Soc. 195 no. 910.
[10]Horton M. D., Stark H. M. and Terras A. A. (2008) Zeta functions of weighted graphs and covering graphs. In Analysis on Graphs and its Applications, Vol. 77 of Proc. Sympos. Pure Math., AMS, Providence, RI, pp. 2950.
[11]Janson S., Łuczak T. and Ruciński A. (2000) Random Graphs, Wiley, New York.
[12]Linial N. and Rozenman E. (2005) Random lifts of graphs: Perfect matchings. Combinatorica 25 407424.
[13]Oren I., Godel A. and Smilansky U. (2009) Trace formulae and spectral statistics for discrete Laplacians on regular graphs I. J. Phys. A: Math. Theor. 42 415101.
[14]McMullen P. (1984) Determinants of lattices induced by rational subspaces. Bull. London Math. Soc. 16 275277.
[15]Molloy M. S. O., Robalewska H., Robinson R. W. and Wormald N. C. (1997) 1-factorizations of random regular graphs. Random Struct. Alg. 10 305321.
[16]Robinson R. W. and Wormald N. C. (1984) Existence of long cycles in random cubic graphs. In Enumeration and Design (Jackson D. M. and Vanstone S. A., eds), Academic Press, Toronto, pp. 251270.
[17]Robinson R. W. and Wormald N. C. (1992) Almost all cubic graphs are Hamiltonian. Random Struct. Alg. 3 117125.
[18]Robinson R. W. and Wormald N. C. (1994) Almost all regular graphs are Hamiltonian. Random Struct. Alg. 5 363374.
[19]Schnell U. (1992) Minimal determinants and lattice inequalities. Bull. London Math. Soc. 24 606612.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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