Taiwanese Journal of Mathematics

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

Daphne Der-Fen Liu

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500404981

Digital Object Identifier
doi:10.11650/twjm/1500404981

Mathematical Reviews number (MathSciNet)
MR2426532

Zentralblatt MATH identifier
1168.05310

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs 11B05: Density, gaps, topology

Keywords
distance graphs fractional chromatic number circular chromatic number density of integral sequences

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


Export citation

References

  • J. Barajas and O. Serra, Distance graphs with maximum chromatic number, Disc. Math., to appear (available on-line).
  • J. Barajas and O. Serra, The lonely runner with seven runners, manuscript.
  • U. Betke and J.M. Wills, Untere Schranken für zwei dophantische approximations-Funktionen, Monatsch. Math., 76 (1972), 214-217.
  • W. Bienia, L. Goddyn, P. Gvozdjak, A. Sebő and M. Tarsi, Flows, view obstructions, and the lonely runner, J. Combin. Theory $($B$)$, 72 (1998), 1-9.
  • T. Bohman, R. Holzman and D. Kleitman, Six lonely runners, Electronic J. Combinatorics, 8 (2001), Research Paper 3, pp. 49.
  • D. G. Cantor and B. Gordon, Sequences of integers with missing differences, J. Combin. Theory (A), 14 (1973), 281-287.
  • G. J. Chang, L. Huang and X. Zhu, The circular chromatic numbers and the fractional chromatic numbers of distance graphs, Europ. J. Combin., 19 (1998), 423-431.
  • G. J. Chang, D. Liu and X. Zhu, Distance graphs and $T$-coloring, J. Combin. Theory (B), 75 (1999), 159-169.
  • J.-J. Chen, G. J. Chang and K.-C. Huang, Integral distance graphs, J. Graph Theory, 25 (1997), 287-294.
  • Y. G. Chen, On a conjecture about Diophantine approximations, II, J. Number Theory, 37 (1991), 181-198.
  • Y. G. Chen, On a conjecture about Diophantine approximations, III, J. Number Theory, 39 (1991), 91-103.
  • Y. G. Chen, On a conjecture about Diophantine approximations, IV, J. Number Theory, 43 (1993), 186-197.
  • T. W. Cusick, View-obstruction problems, Aequationes Math., 9 (1973), 165-170.
  • T. W. Cusick, View-obstruction problems II, Proc. Amer. Math. Soc., 84 (1982), 25-28.
  • T. W. Cusick, View-obstruction problems in $n$-dimensional geometry, J. Combin. Theory (A), 16 (1974), 1-11.
  • T. W. Cusick and C. Pomerance, View-obstruction problems, III, J. Number Theory, 19 (1984), 131-139.
  • R. B. Eggleton, P. Erdős and D. K. Skilton, Colouring the real line, J. Combin. Theory $($B$)$, 39 (1985), 86-100.
  • R. B. Eggleton, P. Erdős and D. K. Skilton, Research problem 77, Disc. Math., 58 (1986), 323.
  • R. B. Eggleton, P. Erdős and D. K. Skilton, Update information on research problem 77, Disc. Math., 69 (1988), 105.
  • R. B. Eggleton, P. Erdős and D. K. Skilton, Colouring prime distance graphs, Graphs and Combin., 6 (1990), 17-32.
  • J. Griggs and D. Liu, The channel assignment problem for mutually adjacent sites, J. Combin. Theory $($A$)$, 68 (1994), 169-183.
  • H. Hadwiger, H. Debrunner and V. Klee, Combinatorial Geometry in the Plane, Holt Rinehart and Winston, New York, 1964.
  • N. M. Haralambis, Sets of integers with missing differences, J. Combin. Theory $($A$)$, 23 (1977), 22-33.
  • L. Huang and G. J. Chang, Circular chromatic number of distance graphs with distance set missing multiples, Europ. J. Combin., 21 (2000), 241-248.
  • A. Kemnitz and H. Kolberg, Coloring of integer distance graphs, Disc. Math., 191 (1998), 113-123.
  • A. Kemnitz and M. Marangio, Colorings and list colorings of integer distance graphs, Congr. Numer., 151 (2001), 75-84.
  • A. Kemnitz and M. Marangio, Chromatic numbers of integer distance graphs, Disc. Math., 233 (2001), 239-246.
  • K.-W. Lih, D. Liu and X. Zhu, Star extremal circulant graphs, SIAM J. Disc. Math., 12 (1999), 491-499.
  • P. Lam and W. Lin, Coloring distance graphs with intervals as distance sets, Europ. J. Combin., 26 (2005), 1216-1229.
  • P. Lam, W. Lin and Z. Song, Circular chromatic numbers of some distance graphs, Disc. Math., 292 (2005), 119-130.
  • W. Lin, Some star extremal circulant graphs, Disc. Math., 271 (2003), 169-177.
  • D. Liu, $T$-coloring and chromatic number of distance graphs, Ars Combin., 56 (2000), 65-80.
  • D. Liu and X. Zhu, Distance graphs with missing multiples in the distance sets, J. Graph Theory, 30 (1999), 245-259.
  • D. Liu and X. Zhu, Fractional chromatic number and circular chromatic number for distance graphs with large clique size, J. Graph Theory, 47 (2004), 129-146.
  • D. Liu and X. Zhu, Fractional chromatic number of distance graphs generated by two-interval sets, Europ. J. Combin., in press, on-line accessible.
  • L. Moser and W. Moser, Solution to problem 10, Canad. Math. Bull., 4 (1961), 187-189.
  • J. H. Rabinowitz and V. K. Proulx, An asymptotic approach to the channel assignment problem, SIAM J. Alg. Disc. Methods, 6 (1985), 507-518.
  • J. Renault, View-obstruction: a shorter proof for 6 lonely runners, Disc. Math., 287 (2004), 93-101.
  • E. R. Scheinerman and D. H. Ullman, Fractional Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, 1997.
  • M. Voigt, Colouring of distance graphs, Ars Combin., 52 (1999), 3-12.
  • M. Voigt and H. Walther, On the chromatic number of a special distance graphs, Disc. Math., 97 (1991), 197-209.
  • M. Voigt and H. Walther, Chromatic number of prime distance graphs, Disc. Appl. Math., 51 (1994), 197-209.
  • H. Walther, Über eine spezielle Klasse unendlicher Graphen, in Graphentheorie II, BI Wissenschaftsverlag, in: K. Wagner und R. Bodendiek (Hrsg.), 1990, 268-295.
  • J. M. Wills, Zwei Sätze über inhomogene diophantische appromixation von irrationlzahlen, Monatsch. Math., 71 (1967), 263-269.
  • J. Wu and W. Lin, Circular chromatic numbers and fractional chromatic numbers of distance graphs with distance sets missing an interval, Ars Combin., 70 (2004), 161-168.
  • X. Zhu, Pattern periodic coloring of distance graphs, J. Combin. Theory $($B$)$, 73 (1998), 195-206.
  • X. Zhu, Circular chromatic number: a survey, Disc. Math., 229 (2001), 371-410.
  • X. Zhu, Recent developments in circular colouring of graphs, Topics in discrete mathematics, 497-550, Algorithms Combin., 26, Springer, Berlin, 2006.
  • X. Zhu, The circular chromatic number of distance graphs with distance sets of cardinality $3$, J. Graph Theory, 41 (2002), 195-207.
  • X. Zhu, The circular chromatic number of a class of distance graphs, Disc. Math., 265 (2003), 337-350.