Discrete Mathematics ( IF 0.770 ) Pub Date : 2020-01-31 , DOI: 10.1016/j.disc.2020.111825
Deborah Oliveros; Christopher O’Neill; Shira Zerbib

An $r$-segment hypergraph $H$ is a hypergraph whose edges consist of $r$ consecutive integer points on line segments in ${\mathbb{R}}^{2}$. In this paper, we bound the chromatic number $\chi \left(H\right)$ and covering number $\tau \left(H\right)$ of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for $r\ge 3$, the covering number $\tau \left(H\right)$ is at most $\left(r-1\right)\nu \left(H\right)$, where $\nu \left(H\right)$ denotes the matching number of $H$. We prove our conjecture in the case where $\nu \left(H\right)=1$, and provide improved (in fact, optimal) bounds on $\tau \left(H\right)$ for $r\le 5$. We also provide sharp bounds on the chromatic number $\chi \left(H\right)$ in terms of $r$, and use them to prove two fractional versions of our conjecture.

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