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
