On the number of Hamiltonian cycles in the generalized Petersen graph

The generalized Petersen graph $G(n, k)$ is a cubic graph with vertex set $V(G(n, k)) = \{v_i\}_{0 \leq i < n} \cup \{w_i\}_{0 \leq i < n}$ and edge set $E(G(n, k)) = \{v_i v_{i+1}\}_{0 \leq i < n} \cup \{w_i w_{i+k}\}_{0 \leq i < n} \cup \{v_i w_i\}_{0 \leq i < n}$ where the indices are taken modulo $n$. Schwenk found the number of Hamiltonian cycles in $G(n, 2)$, and in this article we present initial conditions and linear recurrence relations for the number of Hamiltonian cycles in $G(n, 3)$ and $G(n, 4)$. This is attained by introducing $G'(n, k)$, which is a modified version of $G(n, k)$, and a subset of its subgraphs which we call admissible, and which are partitioned into different classes in such a manner that we can find relations between the number of admissible subgraphs of each class. The classes and their relations define a directed graph such that each strongly connected component is of a manageable size for $k=3$ and $k=4$, which allows us to find linear recurrence relations for the number of admissible subgraphs in each class in these cases. The number of Hamiltonian cycles in $G(n, k)$ is a sum of the number of admissible subgraphs of $G'(n, k)$ over a certain subset of the classes.