Grinberg's Criterion Applied to Some Non-Planar Graphs

Robertson ([5]) and independently, Bondy ([1]) proved that the generalized Petersen graph P(n, 2) is non-hamiltonian if n equivalent to 5 ( mod 6), while Thomason [7] proved that it has precisely 3 hamiltonian cycles if n equivalent to 3 (mod 6). The hamiltonian cycles in the remaining generalized Petersen graphs were enumerated by Schwenk [6]. In this note we give a short unified proof of these results using Grinberg's theorem.