236 research outputs found

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

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    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

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    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

    Trees and Matchings

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    In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph H. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the ``square-octagon'' lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon, our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson's algorithm allows us to quickly generate random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1

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

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    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GG有一个Pfaffian定向,就可以在多项式的时间内计算出GG 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 GG 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
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