Enumeration of Hamiltonian cycles in certain generalized Petersen graphs (Q580371)

The generalized Petersen graph P(n,k) has vertex set \(V=\{u_ 0,u_ 1,...,u_{n-1}\), \(v_ 0,v_ 1,...,v_{n-1}\}\) and edge set \(E=\{u_ iu_{i+1},u_ iv_ i,v_ iv_{i+k}|\) for \(0\leq i\leq n-1\) with indices taken modulo \(n\}\). The classification of the Hamiltonicity of generalized Petersen graphs was begun by Watkins, continued by Bondy and Bannai, and completed by Alspach. We now determine the precise number of Hamiltonian cycles present in each of the graphs P(n,2). This more detailed information allows us to identify an infinite family of counterexamples to a conjecture of Greenwell and Kronk who had suggested a relation between uniquely 3-edge-colorable cubic graphs and the number of Hamiltonian cycles present.