The First Three Levels of an Order Preserving Hamiltonian Path in the Subset Lattice

摘要

Consider the lattice whose elements are the subsets of the set of positive integers not greater than n ordered by inclusion. The Hasse diagram of this lattice is isomorphic to the n-dimensional hypercube. It is trivial that this graph is Hamiltonian. Let {S_1,..., S_2") be a Hamiltonian path. We say it is monotone, if for every i, either (a) all subsets of S_1 appear among S_1,..., S_(i-1), or (b) only one (say S) does not, furthermore S_(i+1) = S. Trotter conjectured that if n is sufficiently large, then there are no monotone Hamiltonian paths in the n-cube. He also made a stronger conjecture that states that there is no path with the monotone property that covers all the sets of size at most three. In this paper we disprove this strong conjecture by explicitly constructing a monotone path covering all the 3-sets.