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

In this lecture we will survey the recent developments in this area and pose several conjectures.

The slides of the lecture are available here.