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 from u to v
in each 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