On the Cartesian product of an arbitrarily partitionable graph and a traceable graph

A graph G of order n is called arbitrarily partitionable (AP, for short) if, for every sequence tau = (n(1), ... , n(k)) of positive integers that sum up to n, there exists a partition (V-1, ... , V-k) of the vertex set V(G) such that each set V-i induces a connected subgraph of order n(i). A graph G is called AP+1 if, given a vertex u is an element of V(G) and an index q is an element of{1, ... , k}, such a partition exists with u is an element of V-q. We consider the Cartesian product of AP graphs. We prove that if G is AP+1 and H is traceable, then the Cartesian product G square H is AP+1. We also prove that G square H is AP, whenever G and H are AP and the order of one of them is not greater than four.