Completely Independent Spanning Trees in Some Regular Networks

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 thevertices u and v, then T1, . . . , Tk are completely independent spanningtrees in G. For 2k-regular graphs which are 2k-connected, such as theCartesian product of a complete graph of order 2k − 1 and a cycle andsome Cartesian products of three cycles (for k = 3), the maximum numberof completely independent spanning trees contained in these graphs isdetermined and it turns out that this maximum is not always k.