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Distance graphs on R n with 1-norm

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Abstract

Suppose S is a subset of a metric space X with metric d. For each subset D⊆{d(x,y):x,y∈S,x≠y}, the distance graph G(S,D) is the graph with vertex set S and edge set E(S,D)={xy:x,y∈S,d(x,y)∈D}. The current paper studies distance graphs on the n-space R n1 with 1-norm. In particular, most attention is paid to the subset Z n1 of all lattice points of R n1 . The results obtained include the degrees of vertices, components, and chromatic numbers of these graphs.

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References

  • Chen JJ (1996) Distance graphs. PhD thesis, Department of Applied Mathematics, National Chiao Tung University

  • Chen JJ, Chang GJ, Huang KC (1997) Integral distance graphs. J Graph Theory 25:287–294

    Article  MATH  Google Scholar 

  • Croft HT, Falconer KJ, Guy RK (1991) Unsolved problems in geometry. Springer, New York

    MATH  Google Scholar 

  • Eggleton RB (1991) Distance graphs and their chromatic numbers. In: Workshop notes of conference of graph theory and combinatorics, Manila

  • Eggleton RB, Erdős P, Skilton DK (1985) Colouring the real line. J Comb Theory Ser B 39:86–100

    Article  Google Scholar 

  • Eggleton RB, Erdős P, Skilton DK (1990) Colouring prime distance graphs. Graphs Comb 6:17–32

    Article  MATH  Google Scholar 

  • Fűredi Z, Kang JH (2004) Distance graphs on Z n with ℓ 1 norm. Theor Comput Sci 319:357–366

    Article  Google Scholar 

  • Fűredi Z, Kang JH (2007) Covering n-space by convex bodies and its chromatic number. Discret Math (to appear)

  • Hadwiger H, Debrunner H, Klee V (1964) Combinatorial geometry in the plane. Rinehart and Winston, New York

    Google Scholar 

  • Jensen TR, Toft B (1995) Graph coloring problems. Wiley, New York

    MATH  Google Scholar 

  • Moser L, Moser W (1961) Solution to problem 10. Can Math Bull 4:187–189

    Google Scholar 

  • Raigorodskii AM (2001) Borsuk’s problem and the chromatic number of some metric spaces. Usp Mat Nauk 56:107–146; translated in Russ Math Surv 56:103–139

    Google Scholar 

  • Voigt M, Walther H (1991) On the chromatic number of special distance graphs. Discret Math 97:395–397

    Article  MATH  Google Scholar 

  • Voigt M, Walther H (1994) Chromatic number of prime distance graphs. Discret Appl Math 51:197–209

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Marketing and Logistics, China University of Technology, Hsinchu, 303, Taiwan

    Jer-Jeong Chen

  2. Department of Mathematics, National Taiwan University, Taipei, 10617, Taiwan

    Gerard J. Chang

Authors
  1. Jer-Jeong Chen
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  2. Gerard J. Chang
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Corresponding author

Correspondence to Jer-Jeong Chen.

Additional information

Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday.

Supported in part by the National Science Council under grant NSC-94-2115-M-002-015. Taida Institue for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office.

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Chen, JJ., Chang, G.J. Distance graphs on R n with 1-norm. J Comb Optim 14, 267–274 (2007). https://doi.org/10.1007/s10878-007-9053-9

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  • DOI: https://doi.org/10.1007/s10878-007-9053-9

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