Let G=(V,E) be an undirected graph with vertices V and edges E. A dimer is a domino occupying an edge e=(u,v) ∈ E and a monomer is a single vertex w ∈ V. An l-match, or a monomer-dimer cover of G, is a subset E' of E, of cardinality l, such that no two distinct edges e,f in E have a common vertex. Let φ(l,G) ≥ 0 be the number of l-matchings in G for any l\in Z+. In many combinatorial problems it is of interest to estimate from above and below the number φ(l,G). In the monomer-dimer models in statistical mechanics, as the integer lattice Zd, or the Bethe lattice, i.e. an infinite k-regular tree, one needs to estimate the number of l-matching in bipartite regular graphs. Moreover, one needs to estimate the corresponding quantities hG(p), called the monomer-dimer entropy, for dimer density p ∈ [0,1], for infinite graphs G.
In this lecture we will survey the recent developments in this area and pose several conjectures.
The slides of the lecture are available here.