## Abstract

We introduce and study a d-dimensional generalization of graph Hamiltonian cycles. These are the Hamiltonian d-dimensional cycles in K_{n}^{d} (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 F_{2} 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(n^{d−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 Z^{d−1}∈K_{n}^{d}, F is its d-filling if ∂F=Z^{d−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 F_{2} 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.

Original language | English |
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Pages (from-to) | 119-143 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 150 |

DOIs | |

State | Published - Sep 2021 |

### Bibliographical note

Funding Information:This Research was supported by The Israel Science Foundation, grant number 497/17.

Publisher Copyright:

© 2021 Elsevier Inc.

## Keywords

- Fillings
- High dimensional trees
- High-dimensional combinatorics
- Hypergraphs Hamiltonian cycles
- Simplicial complexes

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics