Constructing node-independent spanning trees on the line graph of the hypercube by an independent forest scheme

Abstract

Due to the application in reliable communication, reliable broadcasting, secure message distribution, etc., node/edge-independent spanning trees (ISTs) have attracted much attention in the past twenty years. However, node/edge conjecture is still open for networks with node/edge-connectivity ≥ 5. So far, results have been obtained on a lot of special networks, but only a few results are reported on the line graphs of them. Hypercubes play important roles in parallel computing systems, and the line graphs of which have been recently adopted for the architectures of data center networks. Since the line graph of n-dimensional hypercube Qn, L(Qn), is (2n−2)-regular, whether there exist 2n−2 node-ISTs rooted at any node on L(Qn) is an open question. In this paper, we focus on the problem of constructing node-ISTs on L(Qn). Firstly, we point out that L(Qn) can be partitioned into 2n−1 complete graphs. Then, based on the complete graphs and n−1 node-ISTs rooted at 0 on Qn−10, we obtain an “independent forest” containing 2n−2 trees on L(Qn). Furthermore, we present an O(N) time algorithm to construct 2n−2 node-ISTs rooted at node [0, 2n−1] isomorphic to each other on L(Qn) based on the independent forest, where N=n×2n−1 is the number of nodes on L(Qn). In addition, we point out that the 2n−2 node-ISTs on L(Qn) is a new method to prove the node/edge-connectivity and the upper bound of (2n−2)-node/edge-wide-diameter of L(Qn).