On the Pancyclicity of Lexicographic Products

We prove that if G and H are graphs containing at least one edge each, then their lexicographic product G[H] is weakly pancyclic, i. e., it contains a cycle of every length between the length of a shortest cycle and that of a longest one. This supports some conjectures on locally connected graphs and on product graphs. We obtain an analogous result on even cycles in products G[H] that are bipartite. We also investigate toughness conditions on G implying that G[H] is hamiltonian (and hence pancyclic). Supported by the project LN00A056 of the Czech Ministry of Education