Coloring Some Finite Sets in ℝn

• Autorzy: József Balogh , Alexandr Kostochka , Andrei Raigorodskii
• Języki publikacji: EN

Abstrakty

EN

This note relates to bounds on the chromatic number χ(ℝn) of the Euclidean space, which is the minimum number of colors needed to color all the points in ℝn so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in ℝn was introduced showing that . For many years, this bound has been remaining the best known bound for the chromatic numbers of some lowdimensional spaces. Here we prove that and find an exact formula for the chromatic number in the case of n = 2k and n = 2k − 1.

Słowa kluczowe

EN

chromatic number; independence number; distance graph

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 1
• Strony: 25-31

Daty

wydano

2013-03-01

online

2013-04-13

Twórcy

autor

József Balogh

• Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

autor

Alexandr Kostochka

• Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA, Sobolev Institute of Mathematics, Novosibirsk, Russia

autor

Andrei Raigorodskii

• Department of Mechanics and Mathematics, Moscow State University, Leninskie gory, Moscow, 119991, Russia Department of Discrete Mathematics, Moscow Institute of Physics and Technology, Dolgoprudny, Russia

Bibliografia

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• [4] A.B. Kupavskiy and A.M. Raigorodskii, On the chromatic number of R9, J. Math. Sci. 163(6) (2009) 720-731. doi:10.1007/s10958-009-9708-4[Crossref]
• [5] D.G. Larman, A note on the realization of distances within sets in Euclidean space, Comment. Math. Helv. 53 (1978) 529-535. doi:10.1007/BF02566096[Crossref]
• [6] D.G. Larman and C.A. Rogers, The realization of distances within sets in Euclidean space, Mathematika 19 (1972) 1-24. doi:10.1112/S0025579300004903[Crossref]
• [7] N. Pippenger and J. Spencer, Asymptotic behavior of the chromatic index for hypergraphs, J. Combin. Theory (A) 51 (1989) 24-42. doi:10.1016/0097-3165(89)90074-5[Crossref]
• [8] A.M. Raigorodskii, On the chromatic number of a space, Russian Math. Surveys 55 (2000) N2, 351-352. doi:10.1070/RM2000v055n02ABEH000281[Crossref]
• [9] A.M. Raigorodskii, The problems of Borsuk and Grünbaum on lattice polytopes, Izv. Math. 69(3) (2005) 81-108. English transl. Izv. Math. 69(3) (2005) 513-537. doi:10.1070/IM2005v069n03ABEH000537[Crossref]

Identyfikatory

DOI

10.7151/dmgt.1641