The chromatic number of the convex segment disjointness graph

Ruy Fabila-Monroy; David R. Wood

The chromatic number of the convex segment disjointness graph

Combinatorics 2011-05-27 v2 Computational Geometry

Authors: Ruy Fabila-Monroy , David R. Wood

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 $\chi(D_n)$ .

Keywords

chromatic number convex segment disjointness graph geometric graph lower bound point set in convex position

Cite

@article{arxiv.1105.4931,
  title  = {The chromatic number of the convex segment disjointness graph},
  author = {Ruy Fabila-Monroy and David R. Wood},
  journal= {arXiv preprint arXiv:1105.4931},
  year   = {2011}
}

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XIV Spanish Meeting on Computational Geometry Alcal\'a de Henares, Spain, June 27--30, 2011

The chromatic number of the convex segment disjointness graph

Abstract

Keywords

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