Abstract
For κ ≥ 2, let H be a k-uniform hypergraph on n vertices and m edges. The transversal number τ (H) of H is the minimum number of vertices that intersect every edge. We consider the following question: Is τ (H) ≤ n/k + m/6? For κ ≥ 4, we show that the inequality in the question does not always hold. However, the examples we present all satisfy Δ(H) ≥ 4. A natural question is therefore whether τ (H) ≤ n/κ + m/6 holds when Δ(H) ≤ 3. Although we do not know the answer, we prove that the bound holds when Δ(H) ≤ 2, and for that case we characterize the hypergraphs for which equality holds. Furthermore, we prove that the bound holds when κ κ = 2 (with no restriction on the maximum degree), and again there we characterize the hypergraphs for which equality holds. Chvátal and McDiarmid [V. Chvátal, C. McDiarmid, Small transversals in hypergraphs, Combinatorica 12 (1992) 19-26] proved that the bound holds for κ = 3 (again with no restriction on the maximum degree). We characterize the extremal hypergraphs in this case.
Original language | English |
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Pages (from-to) | 959-966 |
Number of pages | 8 |
Journal | Discrete Mathematics |
Volume | 313 |
Issue number | 8 |
DOIs | |
Publication status | Published - 2013 |
Keywords
- Affine plane
- Hypergraph
- Transversal
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics