Hamiltonicity and pancyclicity of cartesian products of graphs

The cartesian product of a graph G with K 2 is called a prism over G . We extend known conditions for hamiltonicity and pancyclicity of the prism over a graph G to the cartesian product of G with paths, cycles, cliques and general graphs. In particular we give results involving cubic graphs and almost claw-free graphs. o prove the following: Let G and H be two connected graphs. Let both G and H have a 2-factor. If Δ ( G ) ≤ g ′ ( H ) and Δ ( H ) ≤ g ′ ( G ) (we denote by g ′ ( F ) the length of a shortest cycle in a 2-factor of a graph F taken over all 2-factorization of F ), then G □ H is hamiltonian.