Independent spanning trees in crossed cubes

Abstract

Multiple independent spanning trees (ISTs) can be used for data broadcasting in networks, which can provide advantageous performances, such as the enhancement of fault-tolerance, bandwidth, and security. However, there is a conjecture on the existence of ISTs in graphs: If a graph G is n-connected (n⩾1), then there are n ISTs rooted at an arbitrary vertex in G. This conjecture has remained open for n⩾5. The n-dimensional crossed cube CQn is a n-connected graph with various desirable properties, which is an important variant of the n-dimensional hypercube. In this paper, we study the existence and construction of ISTs in crossed cubes. We first give a proof of the existence of n ISTs rooted at an arbitrary vertex in CQn(n⩾1). Then, we propose an O(Nlog2N) constructive algorithm, where N=2n is the number of vertices in CQn.