The chromatic number of the convex segment disjointness graph

TL;DR

This paper investigates the chromatic number of a graph formed by segments between points in convex position, providing improved bounds that are nearly tight, advancing understanding of geometric graph coloring.

Contribution

It improves the lower bound on the chromatic number of the convex segment disjointness graph, approaching a tight bound and refining previous estimates.

Findings

Lower bound improved to n - √(2n)

Near-tight bounds established for the chromatic number

Advances understanding of geometric graph coloring in convex configurations

Abstract

Let $P$ be a set of $n$ points in general and convex position in the plane. Let $D_n$ be the graph whose vertex set is the set of all line segments with endpoints in $P$, where disjoint segments are adjacent. The chromatic number of this graph was first studied by Araujo et al. [\emph{CGTA}, 2005]. The previous best bounds are $\frac{3n}{4}\leq\chi(D_n) <n-\sqrt{\frac{n}{2}}$ (ignoring lower order terms). In this paper we improve the lower bound to $\chi(D_n)\geq n-\sqrt{2n}$, to conclude a near-tight bound on $\chi(D_n)$.