Covering the n-space by convex bodies and its chromatic number

Z. Füredi, J. H. Kang

Research output: Article

12 Citations (Scopus)

Abstract

Rogers [A note on coverings, Matematika 4 (1957) 1-6] proved, for a given closed convex body C in n-dimensional Euclidean space Rn, the existence of a covering for Rn by translates of C with density cn ln n for an absolute constant c. A few years later, Erdo{combining double acute accent}s and Rogers [Covering space with convex bodies, Acta Arith. 7 (1962) 281-285] obtained the existence of such a covering having not only low-density cn ln n but also low multiplicity c n ln n for an absolute constant c. In this paper, we give a simple proof of Erdo{combining double acute accent}s and Rogers' theorem using the Lovász Local Lemma. Furthermore, we apply the result to the chromatic number of the unit-distance graph under ℓp-norm.

Original languageEnglish
Pages (from-to)4495-4500
Number of pages6
JournalDiscrete Mathematics
Volume308
Issue number19
DOIs
Publication statusPublished - okt. 6 2008

Fingerprint

Convex Body
Chromatic number
Covering
Acute
Distance Graph
Covering Space
Euclidean space
n-dimensional
Lemma
Multiplicity
Norm
Closed
Unit
Theorem

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Covering the n-space by convex bodies and its chromatic number. / Füredi, Z.; Kang, J. H.

In: Discrete Mathematics, Vol. 308, No. 19, 06.10.2008, p. 4495-4500.

Research output: Article

Füredi, Z. ; Kang, J. H. / Covering the n-space by convex bodies and its chromatic number. In: Discrete Mathematics. 2008 ; Vol. 308, No. 19. pp. 4495-4500.
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