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A Lower Bound for the Radio Number of Graphs

  • Devsi BantvaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

Abstract

A radio labeling of a graph G is a mapping \(\varphi : V(G) \rightarrow \{0, 1, 2,\ldots \}\) such that \(|\varphi (u)-\varphi (v)|\ge \mathrm{diam}(G) + 1 - d(u,v)\) for every pair of distinct vertices uv of G, where \(\mathrm{diam}(G)\) and d(uv) are the diameter of G and distance between u and v in G, respectively. The radio number \(\mathrm{rn}(G)\) of G is the smallest number k such that G has radio labeling with \(\max \{\varphi (v):v \in V(G)\} = k\). In this paper, we slightly improve the lower bound for the radio number of graphs given by Das et al. in [5] and, give necessary and sufficient condition to achieve the lower bound. Using this result, we determine the radio number for cartesian product of paths \(P_{n}\) and the Peterson graph P. We give a short proof for the radio number of cartesian product of paths \(P_{n}\) and complete graphs \(K_{m}\) given by Kim et al. in [6].

Keywords

Radio labeling Radio number Peterson graph Cartesian product of graphs 

Notes

Acknowledgements

I want to express my deep gratitude to anonymous referees for kind comments and constructive suggestions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Lukhdhirji Engineering CollegeMorviIndia

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