Hamiltonian properties in generalized lexicographic products

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 generalize the 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.