On the sizes of $(k,l)$-edge-maximal $r$-uniform hypergraphs

TL;DR

This paper establishes tight bounds on the sizes of $(k,l)$-edge-maximal $r$-uniform hypergraphs, extending previous results and providing a comprehensive understanding of their structural properties.

Contribution

It derives the exact lower and upper bounds for the sizes of $(k,l)$-edge-maximal hypergraphs, demonstrating these bounds are optimal and extending prior work in the field.

Findings

Derived tight bounds for hypergraph sizes

Proved bounds are the best possible

Extended previous theoretical results

Abstract

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$. A $r$-uniform hypergraph $H=(V,E)$ is $(k,l)$-edge-maximal if every subhypergraph $H'$ of $H$ with $|V(H')|\geq l$ has edge-connectivity at most $k$, but for any edge $e\in E(K_n^r)\setminus E(H)$, $H+e$ contains at least one subhypergraph $H''$ with $|V(H'')|\geq l$ and edge-connectivity at least $k+1$. In this paper, we obtain the lower bounds and the upper bounds of the sizes of $(k,l)$-edge-maximal hypergraphs. Furthermore, we show that these bounds are best possible. Thus prior results in [Y.Z. Tian, L.Q. Xu, H.-J. Lai, J.X. Meng, On the sizes of $k$-edge-maximal $r$-uniform hypergraphs, arXiv:1802.08843v3] are extended.