Abstract. We prove a technical theorem about the numbers of crossings in T-joins in different drawings of a fixed graph. As a corollary we characterize Pfaffian graphs in terms of their drawings in the plane and give a new proof of a theorem of Kleitman on the parity of crossings in drawings of K2j+1 and K2j+1,2k+1. This gives a new proof of the Hanani-Tutte theorem. 1.

Abstract. We prove that a graph is Pfaffian if and only if it can be drawn in the plane (possibly with crossings) so that every perfect matching intersects itself an even number of times. 1.

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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...

We consider the problem of computing the permanent of circulant matrices.

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We present a new expression for the partition function of the dimer arrangements and the Ising partition function of the 3-dimensional cubic lattice. We use the Pfaffian method. The partition functions are expressed by means of expectations of determinants and Pfaffians of matrices associated with the cubic lattice.

We study the phase transition in the honeycomb dimer model (equivalently, monotone non-intersecting lattice path model). At the critical point the system has a strong long-range dependence; in particular, periodic boundary conditions give rise to a “resonance” phenomenon, where the partition function and other properties of the system depend sensitively on the shape of the domain.

Abstract — We address the well-known problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a two-dimensional constrained coding system is still an elusive research challenge. The only known exception in the two-dimensional case is an exact (however, not rigorous) solution to the (1, ∞)-RLL system on the hexagonal lattice. Furthermore, only exponential-time algorithms are known for the related problem of counting the exact number of constrained two-dimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a two-dimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graph-theoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique we derive a closed form solution to the capacity related to the Path-Cover constraint in a twodimensional triangular array (the resulting calculated capacity is 0.72399217...). Path-Cover is a generalization of the well known one-dimensional (0, 1)-RLL constraint for which the capacity is known to be 0.69424... Index Terms — capacity of constrained systems, capacity of two-dimensional constrained systems, holographic reductions, networks of relations, FKT method, spectral distribution of Toeplitz matrices I.

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 x1, x2, · · · , xm and y1, y2, · · · , ym, respectively, where xi corresponds to yi(i = 1, 2,..., m). We denote by Qm,n,r, the graph obtained from quadrilateral cylinder of length m and breadth n by adding edges xiyi+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 Qm,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 Qm,n,r by enumerating Pfaffians. Keywords: Pfaffian; Perfect matching; Quadratic lattice; Torus. 1