On the Structure of Middle-levels Graphs

Let 0 < k ∈ Z and let n = 2k + 1. A rooted binary tree T is presented whose nodes are the classes mod n of the weight-k n-tuples over F2. Lexical 1-factorizations of the middle-levels graphs Mk allow to present an equivalent form of T via nodes that are the classes mod n of the weight-(k + 1) n-tuples over F2. This in turn allows to present a quotient graph Rk of Mk via reverse complementation as a possible tool around the Hável conjecture on the existence of Hamilton cycles for all Mk. A presentation of Rk via certain integer (k − 1)-tuples called k-ranges yields a novel viewpoint. The proposal of a natural-ordering and some furthering conjectures conclude the present stage of this work.