# A Lower Bound for the Radio Number of Graphs

## 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 *u*, *v* of *G*, where \(\mathrm{diam}(G)\) and *d*(*u*, *v*) 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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