Abstract
The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function \(f:V(G)\rightarrow \{0,1,2,\ldots ,k\}\) such that \(|f(u)-f(v)|\ge p\) if \(d(u,v)=1\) and \(|f(u)-f(v)|\ge 1\) if \(d(u,v)=2\), where d(u, v) is the distance between the two vertices u and v in the graph. Denote \(\lambda _{p,1}^l(G)= \min \{k \mid G\) has a list k-L(p, 1)-labeling\(\}\). In this paper we show upper bounds \(\lambda _{1,1}^l(G)\le \Delta +9\) and \(\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}\) for planar graphs G without 4- and 6-cycles, where \(\Delta \) is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.
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This work was supported in part by National Natural Science Foundation of China (11501316, 11771403) and Shandong Provincial Natural Science Foundation of China (ZR2017QA010, ZR2017MF055).
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Zhu, J., Bu, Y., Pardalos, M.P. et al. Optimal channel assignment and L(p, 1)-labeling. J Glob Optim 72, 539–552 (2018). https://doi.org/10.1007/s10898-018-0647-9
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DOI: https://doi.org/10.1007/s10898-018-0647-9