A Hamilton Path Heuristic with Applicationsto the Middle Two Levels

The notorious middle two levels problem is to nd a Hamilton cycle in the middle two 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 result, due to Moews and Reid 11] in 1990, was that M 2k+1 is Hamiltonian for all positive k through k = 11. We show that if a Hamilton path between two distinguished vertices exists in a reduced graph then a Hamilton cycle can be constructed in the middle two levels. We describe a heuristic for nding Hamilton paths and apply it to the reduced graph to extend the previous best known results. This also improves the best lower bound on the length of a longest cycle in M 2k+1 for any k.