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Distance Graphs with Finite Chromatic Number

✍ Scribed by I.Z. Ruzsa; Zs. Tuza; M. Voigt

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 96 KB
Volume: 85
Category: Article
ISSN: 0095-8956
DOI: 10.1006/jctb.2001.2093

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✦ Synopsis

The distance graph G(D) with distance set D={d 1 , d 2 , ...} has the set Z of integers as vertex set, with two vertices i, j ¥ Z adjacent if and only if |i -j| ¥ D. We prove that the chromatic number of G(D) is finite whenever inf{d i+1 /d i } > 1 and that every growth speed smaller than this admits a distance set D with infinitechromatic G(D).

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