Hamiltonian and pseudo-Hamiltonian cycles and fillings in simplicial complexes

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