Advertisement

Graphs and Combinatorics

, Volume 25, Issue 2, pp 239–251 | Cite as

An Asymptotic Independence Theorem for the Number of Matchings in Graphs

  • Elmar Teufl
  • Stephan WagnerEmail author
Article

Abstract

Let z(G) be the number of matchings (independent edge subsets) of a graph G. For a set M of edges and/or vertices, the ratio \(r_{G}(M) = z(G \setminus M)/z(G)\) represents the probability that a randomly picked matching of G does not contain an edge or cover a vertex that is an element of M. We provide estimates for the quotient \(r_{G}(A \cup B)/(r_{G}(A)r_G(B))\), depending on the sizes of the disjoint sets A and B, their distance and the maximum degree of the underlying graph G. It turns out that this ratio approaches 1 as the distance of A and B tends to ∞, provided that the size of A and B and the maximum degree are bounded, showing asymptotic independence. We also provide an application of this theorem to an asymptotic enumeration problem related to the dimer-monomer model from statistical physics.

Keywords

Matchings Independence Set distance Asymptotic enumeration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aho, A.V., Sloane, N.J.A.: Some doubly exponential sequences. Fibonacci Quart. 11(4), 429–437 (1973)Google Scholar
  2. 2.
    Bollobás, B., McKay, B.D.: The number of matchings in random regular graphs and bipartite graphs. J. Combin. Theory Ser. B 41(1), 80–91 (1986)Google Scholar
  3. 3.
    Chang, S.C., Chen, L.C.: Dimer-monomer model on the Sierpinski gasket. Physica A: Stat. Mech. Appl. 387(7), 1551–1566 (2008). doi: 10.1016/j.physa.2007.10.057.ArXiv:cond-mat/0702071v1
  4. 4.
    Chowla, S.: The asymptotic behavior of solutions of difference equations. In: Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), vol. I, p. 377. Amer. Math. Soc. (1952)Google Scholar
  5. 5.
    Gaunt, D.S.: Exact Series-Expansion Study of the Monomer-Dimer Problem. Phys. Rev. 179(1), 174–186 (1969)Google Scholar
  6. 6.
    Godsil, C.D., Gutman, I.: On the theory of the matching polynomial. J. Graph Theory 5(2), 137–144 (1981)Google Scholar
  7. 7.
    Gutman, I., Polansky, O.E.: Mathematical concepts in organic chemistry. Springer-Verlag, Berlin (1986)Google Scholar
  8. 8.
    Heilmann, O.J., Lieb, E.H.: Monomers and Dimers. Phys. Rev. Lett. 24(25), 1412–1414 (1970)Google Scholar
  9. 9.
    Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. Comm. Math. Phys. 25, 190–232 (1972)Google Scholar
  10. 10.
    Hosoya, H.: Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons. Bull. Chem. Soc. Jpn. 44, 2332–2339 (1971)Google Scholar
  11. 11.
    Hosoya, H.: Topological index as a common tool for quantum chemistry, statistical mechanics, and graph theory. In: Mathematical and computational concepts in chemistry (Dubrovnik, 1985), Ellis Horwood Ser. Math. Appl., pp. 110–123. Horwood, Chichester (1986)Google Scholar
  12. 12.
    Hou, Y.: On acyclic systems with minimal Hosoya index. Discrete Appl. Math. 119(3), 251–257 (2002)Google Scholar
  13. 13.
    Jerrum, M.: Two-dimensional monomer-dimer systems are computationally intractable. J. Statist. Phys. 48(1–2), 121–134 (1987)Google Scholar
  14. 14.
    Kong, Y.: Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density. Phys. Rev. E (3) 74(1), 061,102, 15 (2006)Google Scholar
  15. 15.
    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com~njas/sequences
  16. 16.
    Teufl, E., Wagner, S.: Enumeration problems for classes of self-similar graphs. J. Combin. Theory Ser. A 114(7), 1254–1277 (2007)Google Scholar
  17. 17.
    Trinajstić, N.: Chemical graph theory, second edn. Mathematical Chemistry Series. CRC Press, Boca Raton, FL (1992)Google Scholar
  18. 18.
    Zhang, L.Z.: The proof of Gutman’s conjectures concerning extremal hexagonal chains. J. Systems Sci. Math. Sci. 18(4), 460–465 (1998)Google Scholar

Copyright information

© Springer Japan 2009

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Department of Mathematical SciencesStellenbosch UniversityMatielandSouth Africa

Personalised recommendations