Constructing two completely independent spanning trees in dual-cubes

Abstract

In this paper, we prove the existence of two completely independent spanning trees in the $n$-dimensional dual-cube $F_n$, a variant of the hypercube, for every $n\ge 5$. To this end, we use the hypercube structure of the clusters of $F_n$ to extend the construction of CIST from the $(n-1)$-dimensional hypercube to the dual-cube. In addition, we provide a recursive algorithm that builds the two trees while improving their diameters. Finally, we propose a conjecture regarding the existence of $k$ completely independent spanning trees in the dual-cube.

arXiv comment24 pages, 9 figures