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The geometry and combinatorics of discrete line segment hypergraphs
Discrete Mathematics ( IF 0.728 ) 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 R2. In this paper, we bound the chromatic number χ(H) and covering number τ(H) of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for r≥3, the covering number τ(H) is at most (r−1)ν(H), where ν(H) denotes the matching number of H. We prove our conjecture in the case where ν(H)=1, and provide improved (in fact, optimal) bounds on τ(H) for r≤5. We also provide sharp bounds on the chromatic number χ(H) in terms of r, and use them to prove two fractional versions of our conjecture.
更新日期:2020-01-31

 

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