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Baolei Cheng, Jianxi Fan, Xiaohua Jia, Shukui Zhang, Bangrui Chen, Constructive Algorithm of Independent Spanning Trees on Möbius Cubes, The Computer Journal, Volume 56, Issue 11, November 2013, Pages 1347–1362, https://doi.org/10.1093/comjnl/bxs123
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Abstract
Independent spanning trees (ISTs) on networks have applications in networks such as reliable communication protocols, the multi-node broadcasting, one-to-all broadcasting, reliable broadcasting and secure message distribution. However, there is a problem on ISTs on graphs: If a graph G is n-connected (n≥1), then there are n ISTs rooted at an arbitrary vertex on G. This problem has remained open for n≥5. In this paper, we consider the construction of ISTs on Möbius cubes—a class of hypercube variants. An O(N log N) recursive algorithm is proposed to construct n ISTs rooted at an arbitrary vertex on the n-dimensional Möbius cube Mn, where N=2n is the number of vertices in Mn. Furthermore, we prove that each IST obtained by our algorithm is isomorphic to an n-level binomial-like tree with the height n+1 for n≥2.