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Saturation for the $3$-uniform loose $3$-cycle

Authors :
English, Sean
Kostochka, Alexandr
Zirlin, Dara
Publication Year :
2022

Abstract

Let $F$ and $H$ be $k$-uniform hypergraphs. We say $H$ is $F$-saturated if $H$ does not contain a subgraph isomorphic to $F$, but $H+e$ does for any hyperedge $e\not\in E(H)$. The saturation number of $F$, denoted $\mathrm{sat}_k(n,F)$, is the minimum number of edges in a $F$-saturated $k$-uniform hypergraph $H$ on $n$ vertices. Let $C_3^{(3)}$ denote the $3$-uniform loose cycle on $3$ edges. In this work, we prove that \[ \left(\frac{4}3+o(1)\right)n\leq \mathrm{sat}_3(n,C_3^{(3)})\leq \frac{3}2n+O(1). \] This is the first non-trivial result on the saturation number for a fixed short hypergraph cycle.<br />Comment: 32 pages, 3 figures

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2202.07149
Document Type :
Working Paper