On the Minimum Size of Hamilton Saturated Hypergraphs

Andrzej Ruciński,Andrzej Żak

doi:10.37236/8414

Abstract

For $1\leq \ell< k$, an $\ell$-overlapping $k$-cycle is a $k$-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two consecutive edges share exactly $\ell$ vertices. A $k$-uniform hypergraph $H$ is $\ell$-hamiltonian saturated if $H$ does not contain an $\ell$-overlapping hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge does contain such a cycle. Let sat$(N,k,\ell)$ be the smallest number of edges in an $\ell$-hamiltonian saturated $k$-uniform hypergraph on $N$ vertices. In the case of graphs Clark and Entringer showed in 1983 that sat$(N,2,1)=\lceil \tfrac{3N}2\rceil$. The present authors proved that for $k\geq 3$ and $\ell=1$, as well as for all $0.8k\leq \ell\leq k-1$, sat$(N,k,\ell)=\Theta(N^{\ell})$. Here we prove that sat$(N,2\ell,\ell)=\Theta\left(N^\ell\right)$.

Highlights

(Aofkv-uerntiifcoersm) ahnydpEerg⊆ra(pVkh)(iks-agrfaapmhilfyorosfhko-retle)misenatpsauibr sHets=of(VV, E), where V called edges is of a H finite set
We will denote by |H| the number of edges in H
In [9] we proved a weaker general upper bound

Summary

Introduction

(Aofkv-uerntiifcoersm) ahnydpEerg⊆ra(pVkh)(iks-agrfaapmhilfyorosfhko-retle)misenatpsauibr sHets=of(VV, E), where V called edges is of a H finite set. Note that the number of edges of an (l, k)-cycle with s vertices is s/(k − l) (and s is divisible by k − l). The largest number of edges in an l-hamiltonian saturated k-graph on N vertices has been determined in [5]. For N divisible by k − l, let sat(N, k, l) be the smallest number of edges in an l-hamiltonian saturated k-graph on N vertices. In the case of graphs, Clark and Entringer proved in 1983 that sat(N, 2, 1). In [9] we proved a weaker general upper bound () k+l sat(N, k, l) = O N 2. In the same paper we(impr)oved the above bound in the smallest open case by showing that sat(N, 4, 2) = O. Our proof combines two general approaches to this type of problems developed, respectively, in [8] and [10, 9]

Parameters setting

Main construction

Proof of Theorem 1

The idea of the proof

Proof of Lemma 6