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

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Abstract

The \(L(3,2,1)\)-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 \(\uplambda _{3,2,1}\)-number of the graph. In this paper, we determine the \(\uplambda _{3,2,1}\)-number for toroidal grids \(T_{n,m}, m \le 6\) and the \(\uplambda _{3,2,1}\)-number for the triangular grid. The last result proves the conjecture from Calamoneri (Inf Process Lett 113:361–364, 2013).

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Authors and Affiliations

  1. School of Information Science and Technology, Chengdu University, Chengdu, 610106, China

    Zehui Shao

  2. Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 , Maribor, Slovenia

    Aleksander Vesel

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  1. Zehui Shao
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  2. Aleksander Vesel
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Correspondence to Aleksander Vesel.

Additional information

Supported by the National Natural Science Foundation of China under the Grant No. 61309015, by the Ministry of Science of Slovenia under the Grant 0101-P-297 and within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation.

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Shao, Z., Vesel, A. \(L(3,2,1)\)-labeling of triangular and toroidal grids. Cent Eur J Oper Res 23, 659–673 (2015). https://doi.org/10.1007/s10100-014-0365-4

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  • DOI: https://doi.org/10.1007/s10100-014-0365-4

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