Let k ≥ 2 be an integer and T1, . . . , T k 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, . . . , T k 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.
A similar work showed that even for 2k-connected and 2k-regular graphs, the maximum number of CISTs is not always k REF .
Completely Independent Spanning Trees in Some Regular Graphs
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Mathematics Computer Science