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28 March 2024
 
  » arxiv » 1710.2501

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On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed
Carlos A. Alfaro ; Adrián Vázquez-Ávila ;
Date 6 Oct 2017
AbstractFor $rgeq2$, let $(P,mathcal{L})$ be an $r$-uniform linear system. The transversal number $ au(P,mathcal{L})$ of $(P,mathcal{L})$ is the minimum number of points that intersect every line of $(P,mathcal{L})$. The 2-packing number $ u_2(P,mathcal{L})$ of $(P,mathcal{L})$ is the maximum number of lines such that the intersection of any three of them is empty. In [Discrete Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it true that if $(P,mathcal{L})$ is a $r$-uniform linear system then $ au(P,mathcal{L})leqdisplaystylefrac{|P|+|mathcal{L}|}{r+1}$ holds for all $kgeq2$?. In this paper, some results about of $r$-uniform linear systems whose 2-packing number is fixed which satisfies the inequality are given.
Source arXiv, 1710.2501
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