Partitioning powers of traceable or hamiltonian graphs

A graph G = (V,E) is arbitrarily partitionable (AP) if for any sequence τ = (n1, . . . , np) of positive integers adding up to the order of G, there is a sequence of mutually disjoints subsets of V whose sizes are given by τ and which induce connected graphs. If, additionally, for given k, it is possible to prescribe l = min{k, p} vertices belonging to the first l subsets of τ , G is said to be AP+k. The paper contains the proofs that the k power of every traceable graph of order at least k is AP+(k − 1) and that the k power of every hamiltonian graph of order at least 2k is AP+(2k−1), and these results are tight.