Kasteleyn's Theorem and Arbitrary Graphs

Abstract

Familiarity with the basic notions of graph theory is assumed. Loops and multiple edges are not permitted. An orientation of an edge e of a graph G is a designation of one of the ends of e as the positive end and the other as the negative end. We say that e is oriented from the positive end to the negative end. If e joins vertex v to vertex w and v is the positive end of e, we write e = (v, w). An orientation of G is a set of orientations, one for each edge of G; a graph with an orientation is called a directed graph.