Matchings in Graphs on Non-orientable Surfaces

We generalize Kasteleynʹs method of enumerating the perfect matchings in a planar graph to graphs embedding on an arbitrary compact boundaryless 2-manifold S. Kasteleyn stated that perfect matchings in a graph embedding on a surface of genus g could be enumerated as a linear combination of 4g Pfaffians of modified adjacency matrices of the graph. We give an explicit construction that not only does this, but also does it for graphs embedding on non-orientable surfaces. If a graph embeds on the connected sum of a genus g surface with a projective plane (respectively, Klein bottle), the number of perfect matchings can be computed as a linear combination of 22g+1 (respectively, 22g+2) Pfaffians. Thus for any S, this is 22−χ(S) Pfaffians. We also introduce “crossing orientations,” the analogue of Kasteleynʹs “admissible orientations” in our context, describing how the Pfaffian of a signed adjacency matrix of a graph gives the sign of each perfect matching according to the number of edge-crossings in the matching. Finally, we count the perfect matchings of an m×n grid on a Möbius strip.