The odd-distance plane graph

Hayri Ardal; Ján Maňuch; Moshe Rosenfeld; Saharon Shelah; Ladislav Stacho

doi:10.1007/s00454-009-9190-2

The odd-distance plane graph

Hayri Ardal, Ján Maňuch, Moshe Rosenfeld^*, Saharon Shelah, Ladislav Stacho

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

Abstract

The vertices of the odd-distance graph are the points of the plane ℝ². Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ² is countably choosable, while such a graph in ℝ³ is not.

Original language	English
Pages (from-to)	132-141
Number of pages	10
Journal	Discrete and Computational Geometry
Volume	42
Issue number	2
DOIs	https://doi.org/10.1007/s00454-009-9190-2
State	Published - Sep 2009

Keywords

Graph coloring
List-chromatic number (choosability)
The unit-distance graph

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10.1007/s00454-009-9190-2

Cite this

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abstract = "The vertices of the odd-distance graph are the points of the plane ℝ2. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ2 is countably choosable, while such a graph in ℝ3 is not.",

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AU - Ardal, Hayri

AU - Maňuch, Ján

AU - Rosenfeld, Moshe

AU - Shelah, Saharon

AU - Stacho, Ladislav

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N2 - The vertices of the odd-distance graph are the points of the plane ℝ2. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ2 is countably choosable, while such a graph in ℝ3 is not.

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KW - Graph coloring

KW - List-chromatic number (choosability)

KW - The unit-distance graph

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JO - Discrete and Computational Geometry

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The odd-distance plane graph

Abstract

Keywords

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