Parallel Construction of Independent Spanning Trees on Enhanced Hypercubes

The use of multiple independent spanning trees (ISTs) for data broadcasting in networks provides a number of advantages, including the increase of fault-tolerance, bandwidth and security. Thus, the designs of multiple ISTs on several classes of networks have been widely investigated. In this paper, we give an algorithm to construct ISTs on enhanced hypercubes $Q_{n,k}$ , which contain folded hypercubes as a subclass. Moreover, we show that these ISTs are near optimal for heights and path lengths. Let $D(Q_{n,k})$ denote the diameter of $Q_{n,k}$ . If $n-k$ is odd or $n-k\in \lbrace 2,n\rbrace$ , we show that all the heights of ISTs are equal to $D(Q_{n,k})+1$ , and thus are optimal. Otherwise, we show that each path from a node to the root in a spanning tree has length at most $D(Q_{n,k})+2$ . In particular, no more than 2.15 percent of nodes have the maximum path length. As a by-product, we improve the upper bound of wide diameter (respectively, fault diameter) of $Q_{n,k}$ from these path lengths.