Lexical tree and linear orderings for the middle-levels graphs

Let m = 2k + 1 ∈ Z be odd. The m-cube graph Hm is the Hasse diagram of the Boolean lattice on the coordinate set Zm. A rooted binary tree T is introduced that has as its nodes the translation classes mod m of the weight-k vertices of all Hm, for 0 < k ∈ Z, with an equivalent form of T whose nodes are the translation classes mod m of weight-(k + 1) vertices via complemented reversals of the former class representatives, both forms of T expressible uniquely via the lexical 1-factorizations of the middle-levels graphs Mk. This yields a linear ordering for both middle levels of any Hm via leftto-right concatenation of the right descending-paths of the size-m nodes of T . The tree T , of interest on its own and whose structure is ruled by Catalan’s triangle, has an inductive definition by means of the said 1-factorizations, which allows to transform each claimed linear ordering into an adjacency list expressible via a 1 k (