A Note on Hamiltonian Cycles in Lexicographical Products

A typical sufficient condition for the existence of a hamiltonian cycle in a lexicographical product G[H] of two graphs G and H forces G to contain a hamiltonian cycle or G to contain a hamiltonian path and H to have some additional properties. We present some sufficient conditions in terms of toughness and factors which are much weaker in many cases. A typical statement is that G[H] is hamiltonian if G is 2-edge connected and cubic and IE] 2 2.