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

Enumeration of perfect matchings of a type of quadratic lattice on the torus

NSFC [10831001]A quadrilateral cylinder of length m and breadth n is the Cartesian product of a m-cycle(with m vertices) and a n-path(with n vertices). Write the vertices of the two cycles on the boundary of the quadrilateral cylinder as x(1), x(2), ... , x(m) and y(1), y(2), ... , y(m), respectively, where x(i) corresponds to y(i) (i = 1, 2, ..., m). We denote by Q(m,n,r), the graph obtained from quadrilateral cylinder of length m and breadth n by adding edges x(i)y(i+r) (r is a integer, 0 <= r < m and i + r is modulo m). Kasteleyn had derived explicit expressions of the number of perfect matchings for Q(m,n,0) [P.W. Kasteleyn, The statistics of dimers on a lattice I: The number of dimer arrangements on a quadratic lattice, Physica 27(1961), 1209-1225]. In this paper, we generalize the result of Kasteleyn, and obtain expressions of the number of perfect matchings for Q(m,n,r) by enumerating Pfaffians

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

From Aztec diamonds to pyramids: steep tilings

We introduce a family of domino tilings that includes tilings of the Aztec diamond and pyramid partitions as special cases. These tilings live in a strip of $\mathbb{Z}^2$ of the form $1 \leq x-y \leq 2\ell$ for some integer $\ell \geq 1$, and are parametrized by a binary word $w\in\{+,-\}^{2\ell}$ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to $w=(+-)^\ell$ and to the limit case $w=+^\infty-^\infty$. For each word $w$ and for different types of boundary conditions, we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips, that admits a natural multivariate generalization. The main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism (which we slightly extend in order to handle Littlewood-type identities). In probabilistic terms our tilings map to Schur processes of different types (standard, Pfaffian and periodic). We also introduce a more general model that interpolates between domino tilings and plane partitions.Comment: 36 pages, 22 figures (v3: final accepted version with new Figure 6, new improved proof of Proposition 11

Enumeration problems on lattices

Thesis (MSc)--Stellenbosch University, 2013.ENGLISH ABSTRACT: The main objective of our study is enumerating spanning trees (G) and perfect matchings PM(G) on graphs G and lattices L. We demonstrate two methods of enumerating spanning trees of any connected graph, namely the matrix-tree theorem and as a special value of the Tutte polynomial T(G; x; y). We present a general method for counting spanning trees on lattices in d 2 dimensions. In particular we apply this method on the following regular lattices with d = 2: rectangular, triangular, honeycomb, kagomé, diced, 9 3 lattice and its dual lattice to derive a explicit formulas for the number of spanning trees of these lattices of finite sizes. Regarding the problem of enumerating of perfect matchings, we prove Cayley’s theorem which relates the Pfaffian of a skew symmetric matrix to its determinant. Using this and defining the Pfaffian orientation on a planar graph, we derive explicit formula for the number of perfect matchings on the following planar lattices; rectangular, honeycomb and triangular. For each of these lattices, we also determine the bulk limit or thermodynamic limit, which is a natural measure of the rate of growth of the number of spanning trees (L) and the number of perfect matchings PM(L). An algorithm is implemented in the computer algebra system SAGE to count the number of spanning trees as well as the number of perfect matchings of the lattices studied.AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is die aftelling van spanbome (G) en volkome afparings PM(G) in grafieke G en roosters L. Ons beskou twee metodes om spanbome in ’n samehangende grafiek af te tel, naamlik deur middel van die matriks-boom-stelling, en as ’n spesiale waarde van die Tutte polinoom T(G; x; y). Ons behandel ’n algemene metode om spanbome in roosters in d 2 dimensies af te tel. In die besonder pas ons hierdie metode toe op die volgende reguliere roosters met d = 2: reghoekig, driehoekig, heuningkoek, kagomé, blokkies, 9 3 rooster en sy duale rooster. Ons bepaal eksplisiete formules vir die aantal spanbome in hierdie roosters van eindige grootte. Wat die aftelling van volkome afparings aanbetref, gee ons ’n bewys van Cayley se stelling wat die Pfaffiaan van ’n skeefsimmetriese matriks met sy determinant verbind. Met behulp van hierdie stelling en Pfaffiaanse oriënterings van planare grafieke bepaal ons eksplisiete formules vir die aantal volkome afparings in die volgende planare roosters: reghoekig, driehoekig, heuningkoek. Vir elk van hierdie roosters word ook die “grootmaat limiet” (of termodinamiese limiet) bepaal, wat ’n natuurlike maat vir die groeitempo van die aantaal spanbome (L) en die aantal volkome afparings PM(L) voorstel. ’n Algoritme is in die rekenaaralgebra-stelsel SAGE geimplementeer om die aantal spanboome asook die aantal volkome afparings in die toepaslike roosters af te tel