Geodesic
On the sizes of $(k,l)$-edge-maximal $r$-uniform hypergraphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 179-194

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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 an 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|. An r-uniform hypergraph H=(V,E) is (k,l)-edge-maximal if every subhypergraph H^' of H with |V(H^')|≥ l has edge-connectivity at most k, but for any edge e∈ E(K_n^r)∖ E(H), H+e contains at least one subhypergraph H^” with |V(H^”)|≥ 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.
Keywords: edge-connectivity, $(k,l)$-edge-maximal hypergraphs, $r$-uniform hypergraphs 
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Tian, Yingzhi; Lai, Hong-Jian; Meng, Jixiang; Xu, Murong. On the sizes of $(k,l)$-edge-maximal $r$-uniform hypergraphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 179-194. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a11/