Counting Structures in Grid Graphs, Cylinders and Tori Using Transfer Matrices: Survey and New Results (Extended Abstract) ∗ Mordecai J. Golin Yiu Cho Leung Yajun Wang Xuerong Yong Abstract There is a very large literature devoted to counting structures, e.g., spanning trees, Hamiltonian cycles, independent sets, acyclic orientations, in the n × m grid graph G(n, m). In particular the problem of counting the number of structures in fixed height graphs, i.e., fixing m and letting n grow, has been, for different types of structures, attacked independently by many different authors, using a transfer matrix approach. This approach essentially permits showing that the number of structures in G(n, m) satisfies a fixed-degree constant-coefficient recurrence relation in n. In contrast there has been surprisingly little work done on counting structures in grid-cylinders (where the left and right, or top and bottom, boundaries of the grid are wrapped around and connected to each other) or in grid-tori (where the left edge of the grid is connected to the right and the top edge is connected to the bottom one). The goal of this paper is to demonstrate that, with some minor modifications, the transfer matrix technique can also be easily used to count structures in fixed height grid-cylinders and tori. 1 Introduction Grid graphs are very common and there is an extremely large literature devoted to counting structures in them. See Table 1. Let G(n, m) denote the n × m grid graph. Much of the counting literature asks questions of the type “let m and/or n go to ∞; how does the number of spanning trees (or Hamiltonian cycles, independent sets, acyclic orientations, k-colorings, etc.) grow as a function of n and/or m. Table 1 presents a selection of these results. Many of the results in this area work by assuming that m (the “height” of the grid) is fixed and examine how the number of structures grows as n →∞; in almost all cases the technique used follows ∗ Partially supported by HK CERG grants HKUST6162/00E, HKUST6082/01E and HKUST6206/02E. The first three authors are at Dept. of Computer Science, Hong Kong U.S.T., Clear Water Bay, Kowloon, Hong Kong. The fourth author is at DIMACS, CoRE Building, Rutgers University, Piscataway, NJ 08854-8018. a transfer matrix formulation (or something equivalent, e.g., recursively calculating the Tutte-polynomial of the growing fixed-height grid [12]). This very natural tech- nique was developed independently by many authors without knowing that it had been used for solving other grid counting problems. The technique permits showing that, for fixed height-m grids, the number of designated structures in G(n, m) will grow as aA n m  b t where A m is some square matrix and a,  b are vectors, all with non- negative integral entries. As will be explained shortly, this immediately implies that, for fixed m, the number of structures in G(n, m) satisfies a fixed-order constant coefficient recurrence relation in n, something that was not a-priori obvious. Given the large amount of prior work on grid-graphs it is surprising to note that there seems to be very little work done on counting structures in related graphs such as cylinders or tori 1 . The main goal of this paper is to show that, by adding a little extra framework, the transfer matrix method can also easily count structures in grid cylinders and tori. We start by formally defining the graphs and the values to be counted. See Figure 1. Definition 1.1. The n × m grid graph Grid Graph G(n, m), has vertex set V (n, m)= {(i, j ):0 ≤ i < n, 0 ≤ j<m} and edge set E G (n, m)= {((i, j ), (i ′ ,j ′ )) : |i − i ′ | + |j − j ′ | =1} . Let Top(n, m) = {((j, m − 1), (j, 0)) : 0 ≤ j<n}, Side(n, m) = {((n − 1,i), (0,i)) : 0 ≤ i<m}. Fat Cylinders FC(n, m), Thin Cylinders TC(n, m) and Tori T (n, m) are graphs with the same vertex set 1 One of the few exceptions is the analysis in [4] of spanning trees in what we will later define as fat-cylinders.