On the Number of Specific Spanning Subgraphs of the Graphs \(G \times F\)

This paper investigates the number of spanning subgraphs of the product of an arbitrary graph \(G\) with the path graphs \(P_n\) 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 \(G \times P_n\), 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 coefficients. 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.