Hamiltonian and pseudo-Hamiltonian cycles and fillings in simplicial complexes

Abstract

We introduce and study a d-dimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian d-dimensional cycles in Knd (the complete simplicial d-complex over a vertex set of size n). Hamiltonian d-cycles are the simple d-cycles of a complete rank, or, equivalently, of size 1+(n−1d).The discussion is restricted to the fields F2 and Q. For d=2, we characterize the n's for which Hamiltonian 2-cycles exist. For d=3 it is shown that Hamiltonian 3-cycles exist for infinitely many n's. In general, it is shown that there always exist simple d-cycles of size (n−1d)−O(nd−3). All the above results are constructive.Our approach naturally extends to (and in fact, involves) d-fillings, generalizing the notion of T-joins in graphs. Given a (d−1)-cycle Zd−1∈Knd, F is its d-filling if ∂F=Zd−1. We call a d-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size (n−1d). If a Hamiltonian d-cycle Z over F2 contains a d-simplex σ, then Z∖σ is a Hamiltonian d-filling of ∂σ (a closely related fact is also true for cycles over Q). Thus, the two notions are closely related.Most of the above results about Hamiltonian d-cycles hold for Hamiltonian d-fillings as well.