On the hamiltonian product of graphs

Let G1 and G2 be graphs and h1, h2 be hamiltonian paths (h-paths) in G1 and G2 respectively. The hamiltonian product (G1, h2)*(G2, h2) was defined by Holton. If a hamiltonian cycle exists in G2, it can give rise to 2n h-paths. Peckham conjectured that (G1, h1)*(G2, h2)≡(G1, h1)*(G2, h3) where h2 and h3 are any two of these 2n h-paths of G2. He has proved the validity of this conjecture for those h2, h3 where h3 is obtainable from h2 by a rotation along the h-cycle of G2. Here we disprove this conjecture for those h2, h3 where one is obtained from the other by a reflection of the h-cycle. © 1976, Australian Mathematical Society. All rights reserved.