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$$L(3,2,1)$$ L ( 3 , 2 , 1 ) -labeling of triangular and toroidal grids

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  • Zehui Shao
  • Aleksander Vesel

Abstract

The $$L(3,2,1)$$ L ( 3 , 2 , 1 ) -labeling of a graph is an abstraction of assigning integer frequencies to radio transceivers such that (1) transceivers that are one unit of distance apart receive frequencies that differ by at least three, (2) transceivers that are two units of distance apart receive frequencies that differ by at least two, and (3) transceivers that are three units of distance apart receive frequencies that differ by at least one. The least span of frequencies in such a labeling is referred to as the $$\uplambda _{3,2,1}$$ λ 3 , 2 , 1 -number of the graph. In this paper, we determine the $$\uplambda _{3,2,1}$$ λ 3 , 2 , 1 -number for toroidal grids $$T_{n,m}, m \le 6$$ T n , m , m ≤ 6 and the $$\uplambda _{3,2,1}$$ λ 3 , 2 , 1 -number for the triangular grid. The last result proves the conjecture from Calamoneri (Inf Process Lett 113:361–364, 2013 ). Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Zehui Shao & Aleksander Vesel, 2015. "$$L(3,2,1)$$ L ( 3 , 2 , 1 ) -labeling of triangular and toroidal grids," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 23(3), pages 659-673, September.
    • Handle: RePEc:spr:cejnor:v:23:y:2015:i:3:p:659-673
    DOI: 10.1007/s10100-014-0365-4
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    References listed on IDEAS

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    1. Smith, D. H. & Hurley, S. & Thiel, S. U., 1998. "Improving heuristics for the frequency assignment problem," European Journal of Operational Research, Elsevier, vol. 107(1), pages 76-86, May.
    2. Dupont, Audrey & Linhares, Andréa Carneiro & Artigues, Christian & Feillet, Dominique & Michelon, Philippe & Vasquez, Michel, 2009. "The dynamic frequency assignment problem," European Journal of Operational Research, Elsevier, vol. 195(1), pages 75-88, May.
    3. Jelena Govorčin & Nebojša Gvozdenović & Janez Povh, 2013. "New heuristics for the vertex coloring problem based on semidefinite programming," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(1), pages 13-25, June.
    4. Tiziana Calamoneri & Emanuele Fusco & Richard Tan & Paola Vocca, 2009. "L(h, 1, 1)-labeling of outerplanar graphs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 69(2), pages 307-321, May.
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