A fully parallelized scheme of constructing independent spanning trees on Mobius cubes

A set of spanning trees in a graph is said to be independent (ISTs for short) if all the trees are rooted at the same node and for any other node , the paths from to in any two trees are node-disjoint except the two end nodes and . It was conjectured that for any -connected graph there exist ISTs rooted at an arbitrary node. Let be the number of nodes in the -dimensional Mobius cube . Recently, for constructing ISTs rooted at an arbitrary node of , Cheng et al. (Comput J 56(11):1347-1362, 2013) and (J Supercomput 65(3):1279-1301, 2013), respectively, proposed a sequential algorithm to run in time and a parallel algorithm that takes time using processors. However, the former algorithm is executed in a recursive fashion and thus is hard to be parallelized. Although the latter algorithm can simultaneously construct ISTs, it is not fully parallelized for the construction of each spanning tree. In this paper, we present a non-recursive and fully parallelized approach to construct ISTs rooted at an arbitrary node of in time using nodes of as processors. In particular, we derive useful properties from the description of paths in ISTs, which make the proof of independency to become easier than ever before.