Abstract
For \(k\ge 2\), 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, we characterize the extremal hypergraphs with equality holds. Based on the proofs, some combinatorial algorithms on the transversal number are designed.
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Acknowledgements
The authors are very indebted to Professor Xujin Chen, Professor Xiaodong Hu and two anonymous referees for their invaluable comments and suggestions.
Funding
This research work is 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) and the Innovation Program for Quantum Science and Technology, China (2021ZD0302904).
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A preliminary version of this paper appeared in Proceedings of the 16th International Conference on Algorithmic Aspects in Information and Management (AAIM 2022), pp. \(376-387, 2022\).
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Chen, Z., Chen, B., Tang, Z. et al. A sharp upper bound for the transversal number of k-uniform connected hypergraphs with given size. J Comb Optim 45, 37 (2023). https://doi.org/10.1007/s10878-022-00968-6
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DOI: https://doi.org/10.1007/s10878-022-00968-6