# Hamiltonian and Pseudo-Hamiltonian Cycles and Fillings In Simplicial Complexes

## TL;DR

This paper generalizes Hamiltonian cycles to higher-dimensional simplicial complexes, characterizes their existence for certain dimensions, and introduces $d$-fillings, extending concepts from graph theory to simplicial complexes.

## Contribution

It introduces and characterizes Hamiltonian $d$-cycles and fillings in simplicial complexes, extending classical graph concepts to higher dimensions with constructive results.

## Key findings

- Hamiltonian 2-cycles characterized for certain $n$
- Existence of Hamiltonian 3-cycles for infinitely many $n$
- Existence of large simple $d$-cycles with size close to maximum

## Abstract

We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} \choose d}$. 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 ${{n-1} \choose d} - O(n^{d-3})$. 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 $Z^{d-1} \in K_n^d$, ~$F$ is its $d$-filling if $\partial F = Z^{d-1}$. We call a $d$-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size ${{n-1} \choose d}$. If a Hamiltonian $d$-cycle $Z$ over $F_2$ contains a $d$-simplex $\sigma$, then $Z\setminus \sigma$ is a a Hamiltonian $d$-filling of $\partial \sigma$ (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.