A sharp upper bound for the edge dominating number of hypergraphs with minimum degree

In a hypergraph H(V, E), a subset of edges \(A\subseteq E\) forms an edge dominating set if each edge \(e\in E\setminus A\) is adjacent to at least one edge in A. The edge dominating number \(\gamma '(H)\) represents the smallest size of an edge dominating set in H. In this paper, we establish upper bounds on the edge dominating number for hypergraphs with minimum degree \(\delta \): (1) For \(\delta \le 4\), \(\gamma '(H)\le \frac{m}{\delta }\); (2) For \(\delta \ge 5\), \(\gamma '(H)\le \frac{m}{\delta }\) holds for hypertrees and uniform hypergraphs; (3) For a random hypergraph model \(\mathcal H(n,m)\), for any positive number \(\varepsilon >0\), \(\gamma ' (H)\le (1+\varepsilon )\frac{m}{\delta }\) holds with high probability when m is bounded by some polynomial function of n. Based on the proofs, some combinatorial algorithms on the edge dominating number of hypergraphs with minimum degree are designed.