A soluble self-avoiding walk problem*)

Synopsis The enumeration of Hamilton walks (complete self-avoiding walks) on a lattice, or close-packed polymer configurations, is discussed. It is shown that if the lines along which the walks proceed are directed (one-way) lines, and the graph formed by the lattice points and these lines is the covering graph of a closed graph, the number of Hamilton walks can be expressed in terms of a determinant and hence calculated explicitly. A quadratic lattice with a certain orientation of the lines between nearest neighbours is treated as an example. By an alternative, but less general, method the generating function for Hamilton walks on this lattice is rigorously calculated as a function of the activities of horizontal and vertical lines.