Independent spanning trees on twisted cubes

Multiple independent spanning trees have applications to fault tolerance and data broadcasting in distributed networks. There are two versions of the n independent spanning trees conjecture. The vertex (edge) conjecture is that any n-connected (n-edge-connected) graph has n vertex-independent spanning trees (edge-independent spanning trees) rooted at an arbitrary vertex. Note that the vertex conjecture implies the edge conjecture. The vertex and edge conjectures have been confirmed only for n-connected graphs with n≤4, and they are still open for arbitrary n-connected graph when n≥5. In this paper, we confirm the vertex conjecture (and hence also the edge conjecture) for the n-dimensional twisted cube TQ_n by providing an O(NlogN) algorithm to construct n vertex-independent spanning trees rooted at any vertex, where N denotes the number of vertices in TQ_n. Moreover, all independent spanning trees rooted at an arbitrary vertex constructed by our construction method are isomorphic and the height of each tree is n+1 for any integer n≥2.