arXiv:1012.0995v2 [math.CO] 27 Jun 2011 How to order the vertices of the middle-levels graphs Italo J. Dejter Abstract. In the absence of a full answer to H´ avel’s conjecture that all middle-levels graphs are hamiltonian, a linear ordering for the vertices of each middle-levels graph can be extracted from a lexical tree containing all vertices of their reduced graphs. This is further applied to determining lower bounds on the abundance of Hamilton cycles in middle-levels graphs, with a short codification, an explicit feature absent from previous papers on the subject. 1. Introduction. If 1 <n ∈ Z, then the n-cube graph H n is defined as the Hasse diagram of the Boolean lattice on the n-element set [n]= {0, 1,...,n - 1}. Vertices of H n will be indicated in three different ways interchangeably: (a) as the subsets A = {a 0 ,a 1 ,...,a r−1 } = a 0 a 1 ...a r−1 of [n] they stand for, where 0 ≤ r ≤ n; (b) as the characteristic n-vectors B A =(b 0 ,b 1 ,...,b n−1 )= b 0 b 1 ...b n−1 over the field F 2 = {0, 1} the subsets A of item (a) represent, given by b i =1 if and only if i ∈ A,(i ∈ [n]; (c) as the polynomials β A (x)= b 0 + b 1 x + ... + b n−1 x n−1 associated to the vectors B A of item (b). We use these three representations interchangeably. A subset A as above is said to be the support of the vector B A . For each k ∈ [n], the k-level L k of H n is the vertex subset of H n formed by those A ⊆ [n] with |A| = k. The middle-levels graph M k is defined as the subgraph of H 2k+1 induced by the union of its k-level and its (k + 1)-level, denoted L k and L k+1 , and formed by the vertices of weights k and k + 1, respectively. Thus, M k is a bipartite graph with vertex parts L k and L k+1 and adjacency given by inclusion, or containment. With the intention of establishing a canonical linear ordering for the vertices of M k , this graph was conjectured to be hamiltonian by I. H´ avel [3], for every 1 <k ∈ Z. The latest partial update on this conjecture is due to I. Shields, B. J. Shields and C. D. Savage [7], who announced the existence of Hamilton cycles in M 16 and M 17 , though their methods do not show explicit presentations, or codifications, of their cycles, as is our case in Section 3 below, with lower bounds on the numbers of such cycles for k ≤ 6. In other directions, J. R. Johnson [5] proved that M k has a cycle of length (1 - o(1)) times the number of vertices, where the term o(1) is of the form c/ √ k, Horak et al. [4] proved that the prism over each M k is hamiltonian, and 1