On properly colored hamiltonian cycles in cubes of Distance-Colored Grids

For a connected graph G and a positive integer k, the kth power Gkof G is the graph with V (Gk) = V (G) where uv ? E(Gk) if the distance dG(u, v) between u and v is at most k. The edge coloring of Gk defined by assigning each edge uv of Gk the color dG(u, v) produces an edge-colored graph Gk called a distance-colored graph. A distance-colored graph Gk is Hamiltonian-colored if Gk contains a properly colored Hamiltonian cycle. For a grid G = Pn Pm with n, m ? 2, it is known that G3 is Hamiltonian-colored and that G2 is Hamiltonian-colored if and only if nm ? 0 (mod 4). It is shown here that (i) G3 contains a properly colored Hamiltonian cycle whose edges are colored only 1 or 3 if and only if nm is even unless n = m = 2 or {n, m} is {2, 3} or {2, 7} and (ii) G3 contains a properly colored Hamiltonian cycle whose edges are colored only 2 or 3 if and only if nm ? 0 (mod 4) unless n = m = 2.