Matchings in Graphs on Non-orientable Surfaces

P.W. Kasteleyn stated that the number of perfect matchings in a graph embedding on a surface of genus g is given by a linear combination of 4 Pfafans of modi ed adjacencymatrices of the graph, but didn't actually give the matrices or the linear combination. We generalize this to enumerating the perfect matchings of a graph embedding on an arbitrary compact boundaryless 2-manifold S with a linear combination of 22 (S) Pfa ans. Our explicit construction proves Kasteleyn's assertion, and additionally treats 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 computedas a linear combinationof 2 (respectively, 2) Pfa ans. We also introduce \crossing orientations," the analogue of Kasteleyn's \admissible orientations" in our context, describing how the Pfa an 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 Mobius strip.