Pfaffian orientation and enumeration of perfect matchings for some Cartesian products of graphs

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. Lovasz 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)) x 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) x G, P(4) x G and P(3) x 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 (G) over right arrow to enumerate their perfect matchings by Pfaffian approach, where (G) over right arrow is an arbitrary orientation of G

The Pfaffian property and enumeration of perfect matchings for some Cartesian product graphs

完美匹配计数问题是匹配理论的一个重要研究内容。 L.Valiant在1979年证明了，一个图（即使是二部图）的完美匹配计数是 NP-hard问题。如果图$G$有一个Pfaffian定向，就可以在多项式的时间内计算出$G$ 的完美匹配个数，所以讨论图的Pfaffian定向具有非常重要的意义。 然而，判断一个图是否具有Pfaffian定向也不是那么容易的。 晏卫根和张福基在文《EnumerationofperfectmatchingsoftypeofCartesianproductsofgraphs，AdvancesinAppl.Math.32(2004)》中 讨论了树与顶点数小于...The enumeration of perfect matchings for a graph is one of important topics of the matching Theory. In 1979, L. Valiant proved that the enumeration of perfect matchings for a graph (even if it is a bipartite graph) is NP-hard. But, if a graph $G$ has a Pfaffian orientation, then the number of its perfect matchings can be calculated in a polynomial time. However, it is not easy to answer tha...学位：理学硕士院系专业：数学科学学院数学与应用数学系_应用数学学号：1912008115275

Combinatorial Potpourri: Permutations, Products, Posets, and Pfaffians

In this dissertation we first examine the descent set polynomial, which is defined in terms of the descent set statistics of the symmetric group. Algebraic and topological tools are used to explain why large classes of cyclotomic polynomials are factors of the descent set polynomial. Next the diamond product of two Eulerian posets is studied, particularly by examining the effect this product has on their cd-indices. A combinatorial interpretation involving weighted lattice paths is introduced to describe the outcome of applying the diamond product operator to two cd-monomials. Then the cd-index is defined for infinite posets, with the calculation of the cd-index of the universal Coxeter group under the Bruhat order as an example. Finally, an extension of the Pfaffian of a skew-symmetric function, called the hyperpfaffian, is given in terms of a signed sum over partitions of n elements into blocks of equal size. Using a sign-reversing involution on a set of weighted, oriented partitions, we prove an extension of Torelli\u27s Pfaffian identity that results from applying the hyperpfaffian to a skew-symmetric polynomial