Counting Structures in Grid Graphs, Cylinders and Tori Using Transfer Matrices: Survey and New Results

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.