Advertisement

# A Lower Bound for the Radio Number of Graphs

• Devsi Bantva
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.

## References

1. 1.
Bantva, D., Vaidya, S., Zhou, S.: Radio number of trees. Electron. Notes Discret. Math. 48, 135–141 (2015)
2. 2.
Bantva, D., Vaidya, S., Zhou, S.: Radio number of trees. Discret. Appl. Math. 217, 110–122 (2017)
3. 3.
Chartrand, G., Erwin, D., Harary, F., Zhang, P.: Radio labelings of graphs. Bull. Inst. Combin. Appl. 33, 77–85 (2001)
4. 4.
Chartrand, G., Erwin, D., Zhang, P.: A graph labeling suggested by FM channel restrictions. Bull. Inst. Combin. Appl. 43, 43–57 (2005)
5. 5.
Das, S., Ghosh, S., Nandi, S., Sen, S.: A lower bound technique for radio $$k$$-coloring. Discret. Math. 340, 855–861 (2017)
6. 6.
Kim, B.M., Hwang, W., Song, B.C.: Radio number for the product of a path and a complete graph. J. Comb. Optim. 30(1), 139–149 (2015)
7. 7.
Liu, D.: Radio number for trees. Discret. Math. 308, 1153–1164 (2008)
8. 8.
West, D.B.: Introduction to Graph Theory. Prentice-Hall of India, New Delhi (2001)Google Scholar

## Copyright information

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

1. 1.Lukhdhirji Engineering CollegeMorviIndia