An update on the middle levels problem

Abstract

The middle levels problem is to find a Hamilton cycle in the middle levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially-ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known, from [I. Shields, C.D. Savage, A Hamilton path heuristic with applications to the middle two levels problem, in: ], was that M 2k+1 is Hamiltonian for all positive k through k = 15. In this note we announce that M 33 and M 35 have Hamilton cycles. The result was achieved by an algorithmic improvement that made it possible to find a Hamilton path in a reduced graph (of complementary necklace pairs) having 129,644,790 vertices, using a 64-bit personal computer.

Synthetic Sequence Design for Signal Location Search

“…Note that Hamiltonian cycles of the middle level graph of B 2k+1 , denoted by M (2k + 1, k), have been verified to exist for k ≤ 19 [27,26]. Furthermore, we use the constructive method shown in the proof of Theorem 1 to obtain the cycle in M (2k + 2, k) spanning the k-set.…”

“…The construction of our 2-consecutive group test design is related to the the middle levels (Revolving Door) conjecture [26]. It is conjectured that there is always a Hamilton cycle in the middle levels, M 2k+1 , of the Hasse diagram of (the partially-ordered set of subsets of a 2k +1-element set ordered by inclusion).…”

“…• Middle-Levels Conjecture Equivalence -The well-known middle levels (or revolving door) conjecture [26] states that the middle levels M 2k+1 of the Hasse diagram of the partial-order defined by inclusion on all subsets of {1, . .…”

Hamiltonian paths in odd graphs

“…Shields and Savage [16] showed that B k has Hamiltonian cycles for 2 ≤ k ≤ 16. Shields, Shields and Savage [18] recently showed that B k also has Hamiltonian cycles for k = 17 and 18. These results [5,6,16,18] were based on properties of smaller graphs obtained by identifying vertices that are equivalent according to some equivalence relation.…”

“…Instead of directly running any heuristics, we use existing results on the middle levels problem [16,18], therefore further relating these two fundamental problems, namely finding a Hamiltonian path in the odd graph and finding a Hamiltonian cycle in the corresponding middle levels graph.…”