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

Z. Füredi, J. H. Kang

Research output: Contribution to journalArticle

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 - Oct 6 2008

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Keywords

  • Chromatic number of the unit-distance graph
  • Convex body
  • Covering
  • Lovász Local Lemma
  • Rogers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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