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 ≡ 5 (mod 6), while Thomason [7] proved that it has precisely 3 hamiltonian cycles if n ≡ 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. A celebrated result of Grinberg (see [4]) concerning planar hamiltonian graphs states that if a planar graph G has a hamiltonian cycle C which partitions its fi faces of degree i into f ′ i (respectively f ′′ i ) faces of degree i in the interior (respectively exterior) of C, then