On the Sizes of Vertex-k-Maximal r-Uniform Hypergraphs

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 hypergraph $H'=(V',E')$ is called a subhypergraph of $H=(V,E)$ if $V'\subseteq V$ and $E'\subseteq E$. A r-uniform hypergraph $H=(V,E)$ is vertex-k-maximal if every subhypergraph of H has vertex-connectivity at most k, but for any edge $e\in E(K_n^r)\setminus E(H)$, $H+e$ contains at least one subhypergraph with vertex-connectivity at least $k+1$. In this paper, we first prove that for given integers n, k, r with $k,r\ge 2$ and $n\ge k+1$, every vertex-k-maximal r-uniform hypergraph H of order n satisfies $|E(H)|\ge (^n_r)-(^{n-k}_r)$, and this lower bound is best possible. Next, we conjecture that for sufficiently large n, every vertex-k-maximal r-uniform hypergraph H on n vertices satisfies $|E(H)|\le (^n_r)-(^{n-k}_r)+(\frac{n}{k}-2)(^k_r)$, where $k,r\ge 2$ are integers. And the conjecture is verified for the case $r>k$.