Constructing sparsest $\ell$-hamiltonian saturated $k$-uniform hypergraphs for a wide range of $\ell$

TL;DR

This paper investigates the minimal size of hypergraphs that are just shy of containing a hamiltonian $( ext{ell},k)$-cycle, extending known results to a new range of parameters and supporting a conjecture about their size.

Contribution

It extends the range of $ ext{ell}$ for which the minimal edge count of saturated hypergraphs is known, confirming the conjecture in the lower-middle range.

Findings

Confirmed the conjecture for $(k-1)/3 \,\leq\, \ell < (k-1)/2$

Extended the validity of the conjecture to a broader parameter range

Provided constructions or bounds for minimal saturated hypergraphs

Abstract

Given $k\ge3$ and $1\leq \ell< k$, an $(\ell,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain hamiltonian $(\ell,k)$-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order $n^\ell$ and confirmed for $\ell\in\{1,k/2,k-1\}$, as well as for the upper range $0.8k\leq \ell\leq k-1$. Here we extend the validity of this conjecture to the lower-middle range $(k-1)/3\le\ell<(k-1)/2$.