The Pfaffian property of Cartesian products of graphs

Suppose that G =( V , E ) is a graph with even vertices. An even cycle C is a nice cycle of G if G − V ( C ) has a perfect matching. An orientation of G is a Pfaffian orientation if each nice cycle C has an odd number of edges directed in either direction of the cycle. Let P n and C n denote the path and the cycle on n vertices, respectively. In this paper, we characterize the Pfaffian property of Cartesian products G × P 2 n and G × C 2 n for any graph G in terms of forbidden subgraphs of G . This extends the results in (Yan and Zhang in Discrete Appl Math 154:145–157, 2006).