Abstract
For \(k\ge 3\), let H be a k-uniform connected hypergraph on n vertices and m edges. The transversal number \(\tau (H)\) is the minimum number of vertices that intersect every edge. We prove the following inequality: \(\tau (H)\le \frac{(k-1)m+1}{k}\). Furthermore, the extremal hypergraphs with equality holds are exactly hypertrees with perfect matching. Based on the proofs, some combinatorial algorithms on the transversal number are designed.
Supported by National Natural Science Foundation of China under Grant No. 11901605, No. 12101069, the disciplinary funding of Central University of Finance and Economics, the Emerging Interdisciplinary Project of CUFE, the Fundamental Research Funds for the Central Universities and Innovation Foundation of BUPT for Youth (500422309).
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Chen, Z., Chen, B., Tang, Z., Diao, Z. (2022). On the Transversal Number of k-Uniform Connected Hypergraphs. In: Ni, Q., Wu, W. (eds) Algorithmic Aspects in Information and Management. AAIM 2022. Lecture Notes in Computer Science, vol 13513. Springer, Cham. https://doi.org/10.1007/978-3-031-16081-3_32
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