## Taiwanese Journal of Mathematics

### FROM RAINBOW TO THE LONELY RUNNER: A SURVEY ON COLORING PARAMETERS OF DISTANCE GRAPHS

Daphne Der-Fen Liu

#### Abstract

Motivated by the plane coloring problem, Eggleton, Erd\H{o}s and Skelton initiated the study of distance graphs. Let $D$ be a set of positive integers. The distance graph generated by $D$, denoted by $G(\mathbb{Z}, D)$, has all integers $\mathbb{Z}$ as the vertex set, and two vertices $x$ and $y$ are adjacent whenever $|x-y| \in D$. The chromatic number, circular chromatic number and fractional chromatic number of distance graphs have been studied extensively in the past two decades; these coloring parameters are also closely related to some problems studied in number theory and geometry. We survey some research advances and open problems on coloring parameters of distance graphs.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 4 (2008), 851-871.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500404981

Digital Object Identifier
doi:10.11650/twjm/1500404981

Mathematical Reviews number (MathSciNet)
MR2426532

Zentralblatt MATH identifier
1168.05310

#### Citation

Liu, Daphne Der-Fen. FROM RAINBOW TO THE LONELY RUNNER: A SURVEY ON COLORING PARAMETERS OF DISTANCE GRAPHS. Taiwanese J. Math. 12 (2008), no. 4, 851--871. doi:10.11650/twjm/1500404981. https://projecteuclid.org/euclid.twjm/1500404981

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