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arXiv · 1804.01057

The exact chromatic number of the convex segment disjointness graph

Ruy Fabila-Monroy·Jakob Jonsson·Pavel Valtr·David R. Wood
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Abstract

Let $P$ be a set of $n$ points in strictly 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, Dumitrescu, Hurtado, Noy, and Urrutia [2005] and then by Dujmović and Wood [2007]. Improving on their estimates, we prove the following exact formula: $$χ(D_n) = n - \left\lfloor \sqrt{2n + \tfrac{1}{4}} - \tfrac{1}{2}\right\rfloor.$$

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Ruy Fabila-Monroy, Jakob Jonsson, Pavel Valtr, David R. Wood. 2018-04-03. The exact chromatic number of the convex segment disjointness graph. https://arxiv.org/abs/1804.01057

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