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

Zehui Shao () and Aleksander Vesel ()

Central European Journal of Operations Research, 2015, vol. 23, issue 3, 659-673

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

Keywords: Frequency assignment; L(3; 2; 1)-labeling triangular grid; Toroidal grid; 05C90; 05C85 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10100-014-0365-4

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