let $H^k$ denote a $k$-dimensional hypercube in a complete symmetric graph $G(V,E)$ without self-loops and parallel edges; let $|V|=2^n$ be the number of vertices.
setting
$\mathbb{H}^0 := V$, i.e. the initial set of $0$-dimensional hypercubes is identical to set $V$ of vertices and
$\mathbb{H}^{k+1} := \left\lbrace H^k_{\pi(2i)}\cup H^k_{\pi(2i+1)}\cup\mathcal{M}\left(V(H^k_{\pi(2i)}),V(H^k_{\pi(2i+1)})\right)\,\Big|\quad 0\le i\le 2^{n-(k+1)}\right\rbrace$, i.e. we combine pairs of $k$-dimensional hypercubes with a perfect bipartite matching between the respective sets of vertices.
Questions:
- does the above algorithm actually generate a spanning hypercube of $G$?
- if not, can we at least say something about the Hamiltonicity of the generated spanner?
