Circular chromatic number of distance graphs with distance sets of cardinality 3

Suppose D is a subset of R+. The distance graph G(R, D) is the graph with vertex set R in which two vertices x,y are adjacent if \x - y\ is an element of D. This study investigates the circular chromatic number and the fractional chromatic number of distance graphs G(R,D) with \D\ = 3. As a consequence, we determine the chromatic numbers of all such distance graphs. This settles a conjecture proposed independently by Chen et al. [J. Chen, G. Chang, and K. Huang, J Graph Theory 25 (1997) 281-287 and Voigt [M. Voigt Ars Combinatoria, 52 (1999) 3-12] in the affirmative. (C) 2002 Wiley Periodicals, Inc.