Pfaffian Orientation and Enumeration of Perfect Matchings for some Cartesian Products of Graphs

Abstract

The importance of Pfaffian orientations stems from the fact that if a graph $G$ is Pfaffian, then the number of perfect matchings of $G$ (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovász pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph $G$ and a cycle $C_n$ with $n$ vertices (or a path $P_n$ with $n$ vertices), we define $C_n$ (or $P_n)\times G$ as the Cartesian product of graphs $C_n$ (or $P_n$) and $G$. In the present paper, we construct Pfaffian orientations of graphs $C_4\times G$, $P_4\times G$ and $P_3\times G$, where $G$ is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of $\overrightarrow{G}$ to enumerate their perfect matchings by Pfaffian approach, where $\overrightarrow{G}$ is an arbitrary orientation of $G$.