Completely independent spanning trees in some regular graphs

Let k≥2 be an integer and T1,…,Tk be spanning trees of a graph G. If for any pair of vertices {u,v} of V(G), the paths between u and v in every Ti, 1≤i≤k, do not contain common edges and common vertices, except the vertices u and v, then T1,…,Tk are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such as the Cartesian product of a complete graph of order 2k−1 and a cycle, and some Cartesian products of three cycles (for k=3), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always k.