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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aho, A.V., Sloane, N.J.A.: Some doubly exponential sequences. Fibonacci Quart. 11(4), 429–437 (1973)
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)
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
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)
Gaunt, D.S.: Exact Series-Expansion Study of the Monomer-Dimer Problem. Phys. Rev. 179(1), 174–186 (1969)
Godsil, C.D., Gutman, I.: On the theory of the matching polynomial. J. Graph Theory 5(2), 137–144 (1981)
Gutman, I., Polansky, O.E.: Mathematical concepts in organic chemistry. Springer-Verlag, Berlin (1986)
Heilmann, O.J., Lieb, E.H.: Monomers and Dimers. Phys. Rev. Lett. 24(25), 1412–1414 (1970)
Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. Comm. Math. Phys. 25, 190–232 (1972)
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)
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)
Hou, Y.: On acyclic systems with minimal Hosoya index. Discrete Appl. Math. 119(3), 251–257 (2002)
Jerrum, M.: Two-dimensional monomer-dimer systems are computationally intractable. J. Statist. Phys. 48(1–2), 121–134 (1987)
Kong, Y.: Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density. Phys. Rev. E (3) 74(1), 061,102, 15 (2006)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com~njas/sequences
Teufl, E., Wagner, S.: Enumeration problems for classes of self-similar graphs. J. Combin. Theory Ser. A 114(7), 1254–1277 (2007)
Trinajstić, N.: Chemical graph theory, second edn. Mathematical Chemistry Series. CRC Press, Boca Raton, FL (1992)
Zhang, L.Z.: The proof of Gutman’s conjectures concerning extremal hexagonal chains. J. Systems Sci. Math. Sci. 18(4), 460–465 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
This material is based upon work supported by the German Research Foundation DFG under grant number 445 SUA-113/25/0-1 and the South African National Research Foundation under grant number 65972.
Rights and permissions
About this article
Cite this article
Teufl, E., Wagner, S. An Asymptotic Independence Theorem for the Number of Matchings in Graphs. Graphs and Combinatorics 25, 239–251 (2009). https://doi.org/10.1007/s00373-008-0832-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-008-0832-6