Hamiltonian-colored powers of strong digraphs

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power D of D is that digraph having vertex set V (D) with the property that (u, v) is an arc of D if the directed distance ~ dD(u, v) from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph D is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph D is distance-colored if each arc (u, v) of D is assigned the color i where i = ~ dD(u, v). The digraph D k is Hamiltonian-colored if D contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which D is Hamiltonian-colored is the Hamiltonian coloring exponent hce(D) of D. For each integer n ≥ 3, the Hamiltonian coloring exponent of the directed cycle ~ Cn of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph Dk such that hce(Dk) = k with the added property that every properly colored Hamiltonian cycle in the kth power of Dk must use all k colors. It is shown for every positive integer p there exists a a connected graph G with two different strong orientations D and D such that hce(D)− hce(D) ≥ p.