Enumeration of matchings in families of self-similar graphs
Discrete Applied Mathematics
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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.