Partitioning powers of traceable or hamiltonian graphs

A graph G=(V,E)G=(V,E) is arbitrarily partitionable (AP) if for any sequence τ=(n1,…,np)τ=(n1,…,np) of positive integers adding up to the order of G, there is a sequence of mutually disjoint 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}l=min{k,p} vertices belonging to the first l subsets of τ, G   is said to be AP+kAP+k.The paper contains the proofs that the kth power of every traceable graph of order at least k   is AP+(k−1)AP+(k−1) and that the kth power of every hamiltonian graph of order at least 2k   is AP+(2k−1)AP+(2k−1), and these results are tight.