Dimer piling problems and interacting field theory

The dimer tiling problem asks in how many ways can the edges of a graph be covered by dimers so that each site is covered once. In the special case of a planar graph, this problem has a solution in terms of a free fermionic field theory. We rediscover and explore an expression for the number of coverings of an arbitrary graph by arbitrary objects in terms of an interacting fermionic field theory first proposed by Samuel. Generalizations of the dimer tiling problem, which we call "dimer piling problems," demand that each site be covered $N$ times by indistinguishable dimers. Our field theory provides a solution of these problems in the large-$N$ limit. We give a similar path integral representation for certain lattice coloring problems.