Constructing two completely independent spanning trees in the dual-cube

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 $n \geq 5$. 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 propose a recursive algorithm that builds the two trees while improving their diameters. Finally, we propose a conjecture concerning the existence of $k$ completely independent spanning trees in the dual-cube.