Abstract
For a simple connected graph G = (V (G),E(G)) and a positive integer k, a radio k-labelling of G is a mapping \(f \colon V(G)\rightarrow \{0,1,2,\ldots \}\) 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 radio k-chromatic number is the minimum span of a radio k-labeling of G. In this article, we study the radio k-labelling problem for complete m-ary trees Tm,h and determine the exact value of radio k-chromatic number for these trees when k ≥ 2h − 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chartrand, G., Erwin, D., Harary, F., Zhang, P.: Radio labelings of graphs. Bull. Inst. Combin. Appl. 33, 77–85 (2001)
Chartrand, G., Erwin, D., Zhang, P: Radio antipodal colorings of cycles. In: Proceedings of the Thirty-first Southeastern International Conference on Combinatorics, Graph Theory and Computing, pp 129–141, Boca Raton FL 2000 (2000)
Das, S., Saha, L., Tiwary, K.: Antipodal radio labelling of full binary trees. In: Zhang, Z., Li, W., Du, D. Z. (eds.) AAIM 2020. LNCS. https://link.springer.com/book/10.1007/978-3-030-57602-8, vol. 12290, pp 456–4689. Springer, Heidelberg (2020)
Das, S., Ghosh, S.C., Nandi, S., Sen, S.: A lower bound technique for radio k-labelling. Discret. Math. 340(5), 855–861 (2017)
Hale, W.K.: Frequency assignment, theory and application. Proc. IEEE 68, 1497–1514 (1980)
Juan, J.S.-T., Liu, D.D.-F.: Antipodal labelings for cycles. Ars Combin. 103, 81–96 (2012)
Khennoufa, R., Togni, O.: The radio antipodal and radio numbers of the hypercube. Ars Combin. 102, 447–461 (2011)
Khennoufa, R., Togni, O.: A note on radio antipodal colourings of paths. Math. Bohem. 130(3), 277–282 (2005)
Liu, D. D.-F.: Radio number for trees. Discret. Math. 308, 1153–1164 (2008)
Liu, D.D.-F., Zhu, X.: Multi-level distance labelings for paths and cycles. SIAM J. Discret. Math. 19, 610–621 (2005)
Liu, D.D.-F., Xie, M.: Radio number for square paths. Ars Combin. 90, 307–319 (2009)
Liu, D.D.-F., Saha, L., Das, S.: Improved lower bounds for the radio number of trees. Theor. Comput. Sci. 851, 1–13 (2021)
Liu, D.D.-F., Xie, M.: Radio number for square of cycles. Congr. Numer. 169, 105–125 (2004)
Li, X., Mak, V., Zhou, S.: Optimal radio labellings of complete m-ary trees. Discret. Appl. Math. 158, 507–515 (2010)
Morris-Rivera, M., Tomova, M., Wyels, C., Yeager, Y.: The radio number of \(C_{n}\Box C_{n}\). Ars Combin. 120, 7–21 (2015)
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)
Reddy, V.S., Viswanathan, P., Iyer, K.: Upper bounds on the radio number of some trees. Int. J. Pure Appl. Math. 71(2), 207–215 (2011)
Saha, L., Panigrahi, P.: On the Radio number of Toroidal grids. Aust. J. Combin. 55, 273–288 (2013)
Saha, L., Panigrahi, P.: A lower bound for radio k-chromatic number. Discret. Appl. Math. 192, 87–100 (2015)
Saha, L., Panigrahi, P.: A Graph Radio k-labelling Algorithm. Lect. Notes Comput. Sci. 7643, 125–129 (2012)
Basunia, R.A., Das, S., Saha, L., Tiwary, K.: Antipodal number of full m-ary trees. Theor. Comput. Sci. https://doi.org/10.1016/j.tcs..2021.06.034
Saha, L., Panigrahi, P., Kumar, P.: On radio number of power of cycles. Asian-Eur. J. Math. 4(3), 523–544 (2011)
Saha, L., Panigrahi, P.: On the radio number of square of graphs. Electron. Notes Discret. Math. 48, 205–212 (2015)
Saha, L.: Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set. AKCE Int. J. Graphs Comb. 17(1), 342–355 (2020)
Saha, L., Das, S., Das, K., Tiwary, K.: Radio k-labeling of Paths. Hacet. J. Math. Stat. 49(6), 1926–1943 (2020)
Zhang, P.: Radio labellings of cycles. Ars Combin. 65, 21–32 (2002)
Acknowledgements
The authors are deeply grateful to the two anonymous referees for their carefulreading of the manuscript and for their helpful and insightful comments. The first author is also thankful to the National Board for Higher Mathematics (NBHM), India, for its financial support (Grant No. 2/48(22)/R & D II/4033).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Saha, L., Basunia, A.R., Das, S. et al. Radio k-chromatic Number of Full m-ary Trees. Theory Comput Syst 66, 114–142 (2022). https://doi.org/10.1007/s00224-021-10056-7
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-021-10056-7