ℓ-distant Hamiltonian walks in Cartesian product graphs

We introduce and study a generalisation of Hamiltonian cycles: an ℓ-distant Hamiltonian walk in a graph G of order n is a cyclic ordering of its vertices in which consecutive vertices are at distance ℓ. Conditions for a Cartesian product graph to possess such an ℓ-distant Hamiltonian walk are given and more specific results are presented concerning toroidal grids.