Enumeration of Perfect Matchings of the Cartesian Products of Graphs

A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G \times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G \times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $G\times P_{2n+1}$ ($n\geqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G \times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G \times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G \times P_{n}$ by the eigenvalues of $G$.