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


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.


Matchings Independence Set distance Asymptotic enumeration 


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© Springer Japan 2009

Authors and Affiliations

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

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