  The lexicographic product $G[H]$ of two graphs $G$ and $H$ is obtained from
$G$ by replacing each vertex with a copy of $H$ and adding all edges between
any pair of copies corresponding to adjacent vertices of $G$. We consider also
the generalized lexicographic product such that we replace each vertex of $G$
with arbitrary graph on the same number of vertices. We present sufficient and
necessary conditions for traceability, hamiltonicity and hamiltonian
connectivity of $G[H]$ if $G$ is a path and hence we improved and extended
results in M. Kriesell, A Note on Hamiltonian Cycles in Lexicographical
Products.
