Independent spanning trees on twisted cubes

Abstract

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 T Q n by providing an O ( N log N ) algorithm to construct n vertex-independent spanning trees rooted at any vertex, where N denotes the number of vertices in T Q 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 .