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.
Alon, N.: Restricted colorings of graphs. In: Proc. 14th British Combinatorial Conference. Surveys in Combinatorics, London Mathematical Society Lecture Notes Series, vol. 187, pp. 1–33. Cambridge University Press, Cambridge (1993)
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry, pp. 234–244. Springer, Berlin (2005)
Erdős, P.: On sets of distances of n points. Am. Math. Mon. 53, 248–250 (1946)
Erdős, P.: Twenty five years of questions and answers. In: Proc. 25th South-Eastern International Conference on Combinatorics, Graph Theory and Computing (1994)
Guldan, F.: On a problem of colouring the real plane. Math. Bohem. 116, 309–318 (1991)
Graham, R., Rothschild, B., Strauss, E.: Are there n+2 points in E n with odd integral distances? Am. Math. Mon. 81, 21–25 (1974)
Maehara, H., Ota, K., Tokushige, N.: Every graph is an integral distance graph in the plane. J. Comb. Theory Ser. A 80, 290–294 (1997)
Piepmeyer, L.: The maximum number of odd integral distances between points in the plane. Discrete Comput. Geom. 16, 113–115 (1996)
Rosenfeld, M.: Odd integral distances among point in the plane. Geombinatorics 5, 156–159 (1996)
Soifer, A.: Chromatic number of the plane & its relatives. Part I: the problem & its history. Geombinatorics 12, 131–148 (2003)
Woodall, D.R.: Distance realized by sets covering the plane. J. Comb. Theory Ser. A 14, 187–200 (1973)
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.
About this article
Cite this article
Ardal, H., Maňuch, J., Rosenfeld, M. et al. The Odd-Distance Plane Graph. Discrete Comput Geom 42, 132–141 (2009). https://doi.org/10.1007/s00454-009-9190-2
- The unit-distance graph
- Graph coloring
- List-chromatic number (choosability)