On the number of specific spanning subgraphs of the graphs G x Pn

This paper investigates the number of spanning subgraphs of the product of an arbitrary graph G with the path graphs Pn on n vertices that meet certain properties: connectivity, acyclicity, Hamiltonicity, and restrictions on degree. A general method is presented for constructing a recurrence equation R(n) for the graphs G Pn, giving the number of spanning subgraphs that satisfy a given combination of the properties. The primary result is that all constructed recurrence equations are homogeneous linear recurrence equations with integer coeecients. A second result is that the property \having a spanning tree with degree restricted to 1 and 3" is a comparatively strong property, just like the property \having a Hamilton cycle", which has been studied extensively in literature.