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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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