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.

A Characterisation of Pfaffian Near Bipartite Graphs

“…The former property implies that the latter is independent of the orientation of H. (See Lemma 2.8.) The following lemma is proved in [2]. Lemma 2.2.…”

“…Let P be a directed A _ B-arc. Then there exist i, j # [1,2] such that P is included in a minimal path Q directed from y i to x j .…”

“…Let i, j # [1,2] and let P be a directed path from y i to x j under our reference orientation. We say that P is minimal if for every edge e # P&(A _ B) there is no directed path from…”

Operations preserving the pfaffian property of a graph

“…(We write (u, w) for an edge of G* directed from vertex u to vertex w .) Associate with f t a plus sign if [2] Pfaffian property 249 and a minus sign otherwise. Thus the signs of the 1-factors are dependent on the choice of / , , but it is clear that the resulting partition of F into two complementary subsets is not.…”

A survey of Pfaffian orientations of graphs

“…A special case is implicit in the work of Fisher [16] and Temperley and Fisher [55]. Different proofs may be found in [25], [26], [32], [38].…”