# Updating the hamiltonian problem

As for the Boolean graphs, B 1 , B 2 and B 3 are easily seen to have a hamiltonian decomposition, while B 4 was shown to have such a decomposition in =-=-=-. An n-bit binary Gray code is an enumeration of all n-bit binary strings so that successive elements differ in exactly one bit position; equivalently, a hamilton path in the Hasse diagram of B n (the partially ordered set of subsets of an n-element set, ordered by inclusion.) We construct, for each n ..." An n-bit binary Gray code is an enumeration of all n-bit binary strings so that successive elements differ in exactly one bit position; equivalently, a hamilton path in the Hasse diagram of B n (the partially ordered set of subsets of an n-element set, ordered by inclusion.) We construct, for each n, a hamilton path in B n with the following additional property: edges between levels i \Gamma 1 and i of B n must appear on the path before edges between levels i and i 1.The Boolean graphs B 5 , B 6 , B 7 and B 8 were all shown to be hamiltonian in ; while independently Dejter  showed B 8 and B 9 were hamiltonian. Two consequences are an embedding of the hypercube into a linear array which simultaneously minimizes dilation in both directions, and a long path in the middle two levels of B n . 2k and [n] = f1; 2; : : : ; ng, let the bipartite graph M n;k have vertices f A [n] j Aj = k or n kg and edges f(A; B) A Bg.We describe a heuristic for finding Hamilton paths and apply it to the reduced graph to extend the previous best known results.

In this note we announce that M33 and M35 have Hamilton cycles.

This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved. There are several open problems about paths between levels of the Hasse diagram of 14 the Boolean lattice, B n .

In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems. The most notorious is the middle two levels problem which is attributed in =-=[KT88]-=- to Dejter, Erdos, and Trotter and by others to H'avel and Kelley.

The middle levels problem remains open, in spite of the efforts of many =-=[3,4,5,6,7,8]-=-.

In 1990, in unpublished work, Moews and Reid verified that M2k 1 is Hamiltonian for 1 ≤ k ≤ 11. In this note we announce that M33 and M35 are Hamiltoni... An intriguing open question is whether the graph formed by the middle two levels of the Boolean lattice of subsets of a 2k 1-element set has a Hamilton path for all k 1: We consider finding a lower bound on the length of the longest cycle in this graph.