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The Odd-Distance Plane Graph


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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Author information

Correspondence to Moshe Rosenfeld.

Additional information

The research of J. Maňuch was supported in part by MITACS (Mathematics of Information Technology and Complex Systems).

The research of M. Rosenfeld was supported in part by the Chancellor Research Grant and the Institute of Technology, UWT.

The research of S. Shelah was supported by the United States-Israel Binational Science Foundation (Grant no. 2002323), and by NSF grant No. NSF-DMS 0600940. No. 923 on Shelah’s publication list.

The research of L. Stacho was supported in part by NSERC (Natural Science and Engineering Research Council of Canada) grant.

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Ardal, H., Maňuch, J., Rosenfeld, M. et al. The Odd-Distance Plane Graph. Discrete Comput Geom 42, 132–141 (2009).

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