Independent spanning trees in crossed cubes

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node r and for any other node v(6= r), the paths from v to r in any two trees are nodedisjoint except the two end nodes v and r. For an n-connected graph, the independent spanning trees problem asks to construct n ISTs rooted at an arbitrary node of the graph. Recently, Zhang et al. [Y.H. Zhang, W. Hao, and T. Xiang, Independent spanning trees in crossed cubes, Inform. Process. Lett., 113 (2013) 653–658] proposed an algorithm to construct n ISTs with a common root at node 0 in an n-dimensional crossed cube CQn. However, it has been proved by Kulasinghe and Bettayeb [P.D. Kulasinghe and S. Bettayeb, Multiplytwisted hypercube with 5 or more dimensions is not vertex transitive, Inform. Process. Lett., 53 (1995) 33–36] that the CQn (a synonym called multiplytwisted hypercube in that paper) fails to be nodetransitive for n > 5. Thus, the result of Zhang et al. does not really solve the ISTs problem in CQn. In this paper, we revisit the problem of constructing n ISTs rooted at an arbitrary node in CQn. As a consequence, we show that the proposed algorithm can be parallelized to run in O(logN) time using N = 2 nodes of CQn as processors. Keyword: independent spanning trees; interconnection networks; crossed cubes; multiply-twisted hypercube; ∗This research was partially supported by National Science Council under the Grants NSC102-2221-E-141-002 and NSC102-2221-E-141-001-MY3. †Corresponding author. Email: [email protected]