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

Zhongzheng Tang () and Zhuo Diao ()
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Zhongzheng Tang: Beijing University of Posts and Telecommunications
Zhuo Diao: Central University of Finance and Economics

Journal of Combinatorial Optimization, 2025, vol. 49, issue 3, No 11, 16 pages

Abstract: Abstract In a hypergraph H(V, E), a subset of edges $$A\subseteq E$$ A ⊆ E forms an edge dominating set if each edge $$e\in E\setminus A$$ e ∈ E \ A is adjacent to at least one edge in A. The edge dominating number $$\gamma '(H)$$ γ ′ ( 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 $$\delta $$ δ : (1) For $$\delta \le 4$$ δ ≤ 4 , $$\gamma '(H)\le \frac{m}{\delta }$$ γ ′ ( H ) ≤ m δ ; (2) For $$\delta \ge 5$$ δ ≥ 5 , $$\gamma '(H)\le \frac{m}{\delta }$$ γ ′ ( H ) ≤ m δ holds for hypertrees and uniform hypergraphs; (3) For a random hypergraph model $$\mathcal H(n,m)$$ H ( n , m ) , for any positive number $$\varepsilon >0$$ ε > 0 , $$\gamma ' (H)\le (1+\varepsilon )\frac{m}{\delta }$$ γ ′ ( H ) ≤ ( 1 + ε ) m δ 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.

Keywords: Edge dominating number; Minimum degree; Upper bound (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10878-025-01284-5

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