Abstract
A multilevel distance labeling of a graph \(G=(V,E)\) is a function \(f\) on \(V\) into \(\mathbb N \cup \{0\}\) such that \(|f(v)-f(w)| \ge \text{ diam }(G)+1-\text{ dist }(v,w)\) for all \(v,w\in V\). The radio number \(\text{ rn }(G)\) of \(G\) is the minimum span over all multilevel distance labelings of \(G\). In this paper, we completely determine the radio number \(\text{ rn }(G)\) of \(G\) where \(G\) is the Cartesian product of a path \(P_n\) with \(n\,(n\ge 4)\) vertices and a complete graph \(K_m\) with \(m\,(m\ge 3)\) vertices.
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Acknowledgments
This work was supported by the Research Institute of Natural Science of Gangneng-Wonju National University. The authors thank the referees for corrections and suggestions.
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Kim, B.M., Hwang, W. & Song, B.C. Radio number for the product of a path and a complete graph. J Comb Optim 30, 139–149 (2015). https://doi.org/10.1007/s10878-013-9639-3
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DOI: https://doi.org/10.1007/s10878-013-9639-3