IWOCA 2012: Combinatorial Algorithms pp 125-129

# A Graph Radio k-Coloring Algorithm

• Laxman Saha
• Pratima Panigrahi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7643)

## Abstract

For a positive integer k, a radio k-coloring of a simple connected graph G = (V(G), E(G)) is a mapping f  : V(G) → { 0,1,2,…} such that $$|f(u)-f(v)|\geqslant k+1-d(u,v)$$ for each pair of distinct vertices u and v of G, where d(u,v) is the distance between u and v in G. The span of a radio k-coloring f, rc k (f), is the maximum integer assigned by it to some vertex of G. The radio k-chromatic number, rc k (G) of G is $$\displaystyle\min\{rc_{k}(f)\}$$, where the minimum is taken over all radio k-colorings f of G. If k is the diameter of G, then rc k (G) is known as the radio number of G. In this paper, we give an algorithm to find an upper bound of rc k (G). We also give an algorithm that implement the result in [16,17] for lower bound of rc k (G). We check that for cycle C n , upper and lower bound obtained from these algorithms coincide with the exact value of radio number, when n is an even integer with $$4\leqslant n\leqslant 400$$. Also applying these algorithms we get the exact value of the radio number of several circulant graphs.

## References

1. 1.
Chartrand, G., Erwin, D., Harrary, F., Zhang, P.: Radio labeling of graphs. Bull. Inst. Combin. Appl. 33, 77–85 (2001)
2. 2.
Chartrand, G., Erwin, D., Zhang, P.: A graph labeling problem suggested by FM channel restrictions. Bull. Inst. Combin. Appl. 43, 43–57 (2005)
3. 3.
Hale, W.K.: Frequency assignment, Theory and application. Proc. IEEE 68, 1497–1514 (1980)
4. 4.
Khennoufa, R., Togni, O.: The radio antipodal and radio numbers of the hypercube. Ars Combin. 102, 447–461 (2011)
5. 5.
Khennoufa, R., Togni, O.: A note on radio antipodal colorigs of paths. Math. Bohem. 130(1), 277–282 (2005)
6. 6.
Kola, S.R., Panigrahi, P.: Nearly antipodal chromatic number ac′(P n) of a path P n. Math. Bohem. 134(1), 77–86 (2009)
7. 7.
Kola, S.R., Panigrahi, P.: On Radio (n − 4)-chromatic number the path P n. AKCE Int. J. Graphs Combin. 6(1), 209–217 (2009)
8. 8.
Kola, S.R., Panigrahi, P.: An improved Lower bound for the radio k-chromatic number of the Hypercube Q n. Comput. Math. Appl. 60(7), 2131–2140 (2010)
9. 9.
Liu, D.D.-F.: Radio number for trees. Discrete Math. 308, 1153–1164 (2008)
10. 10.
Liu, D.D.-F., Xie, M.: Radio Number for Square Paths. Ars Combin. 90, 307–319 (2009)
11. 11.
Liu, D.D.-F., Xie, M.: Radio number for square of cycles. Congr. Numer. 169, 105–125 (2004)
12. 12.
Liu, D., Zhu, X.: Multi-level distance labelings for paths and cycles. SIAM J. Discrete Math. 19(3), 610–621 (2005)
13. 13.
Li, X., Mak, V., Zhou, S.: Optimal radio labellings of complete m-ary trees. Discrete Appl. Math. 158, 507–515 (2010)
14. 14.
Morris-Rivera, M., Tomova, M., Wyels, C., Yeager, Y.: The radio number of C nDC n. Ars Combin. (to appear)Google Scholar
15. 15.
Ortiz, J.P., Martinez, P., Tomova, M., Wyels, C.: Radio numbers of some generalized prism graphs. Discuss. Math. Graph Theory 31(1), 45–62 (2011)
16. 16.
Saha, L., Panigrahi, P.: Antipodal number of some powers of cycles. Discrete Math. 312, 1550–1557 (2012)
17. 17.
Saha, L., Panigrahi, P.: On Radio number of power of cycles. Asian-European J. Math. 4, 523–544 (2011)