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, the number of perfect matchings of G (as well as other related problems) can be computed polynomial time. Although there are many equivalent conditions for the existence a Pfaffian orientation of a graph, this property is not well-characterized. The problem is no polynomial algorithm is known for checking whether or not a given orientation of a is Pfaffian. Similarly, we do not know whether this property of an undirected graph it has a Pfaffian orientation is in NP. It is well known that the enumeration problem perfect matchings for general graphs is NP-hard. L. Lov́asz pointed out that it makes not only to seek good upper and lower bounds of the number of perfect matchings general graphs, but also to seek special classes for which the problem can be solved . For a simple graph G and a cycle Cn with n vertices (or a path Pn with n vertices), define Cn (or P n)×excl;¿G as the Cartesian product of graphs C n (or Pn) and G. In the present , we construct Pfaffian orientations of graphs C4 ×excl;¿ G, P4 ×excl;¿ G and P3 ×excl;¿ G, where G is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of of the skew adjacency matrix of -?excl;æG to enumerate their perfect matchings by Pfaffian approach, where excl;æG is an arbitrary orientation of G.