The geometry and combinatorics of discrete line segment hypergraphs

TL;DR

This paper investigates the geometric and combinatorial properties of discrete line segment hypergraphs, providing bounds on their chromatic and covering numbers, and proposing conjectures supported by proofs and optimal bounds for specific cases.

Contribution

It introduces bounds on chromatic and covering numbers of r-segment hypergraphs, proves a conjecture for certain cases, and explores fractional versions of the conjecture.

Findings

Bounded the chromatic number $ ext{χ}(H)$ in terms of r.

Established the covering number $ au(H)$ bounds for r ≤ 5.

Proved the conjecture for hypergraphs with matching number $ u(H)=1$.

Abstract

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 $\chi(H)$ and covering number $\tau(H)$ of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for $r \ge 3$, the covering number $\tau(H)$ is at most $(r - 1)\nu(H)$, where $\nu(H)$ denotes the matching number of $H$. We prove our conjecture in the case where $\nu(H) = 1$, and provide improved (in fact, optimal) bounds on $\tau(H)$ for $r \le 5$. We also provide sharp bounds on the chromatic number $\chi(H)$ in terms of $r$, and use them to prove two fractional versions of our conjecture.