Radio Number

Let  be a finite, connected, undirected graph with graph diameter  and graph distance  between vertices  and . A radio labeling of a graph  is labeling using distinct nonnegative integers such that  for every pair of distinct vertices ,  in the vertex set of . Then the radio number of , commonly denoted , is the smallest integer  such that  has a radio labeling  with .

The radio number of path graphs  and cycle graphs  were determined by Liu and Zhu (2005). The following table summarizes some known results for a number of special families of graphs.

graph	radio number
complete graph
complete bipartite graph
complete tripartite graph
crown graph	for
cycle graph
path graph
star graph
wheel graph

REFERENCES

Bantva, D. "Further Results on the Radio Number of Trees." 25 May 2018. https://arxiv.org/abs/1805.10083.

Chartrand, G.; Erwin, D.; Harary, F.; and Zhang, P. "Radio Labelings of Graphs." Bull. Inst. Combin. Appl. 33, 77-85, 2001.

Chartrand G.; and Zhang, P. "Radio Colorings of Graphs--A Survey." Int. J. Comput. Appl. Math. 2, 237-252, 2007.

Griggs, J. R. and Yeh, R. K. "Labeling Graphs with Condition at Distance. 2." SIAM J. Disc. Math. 5, 586-595, 1992.

Liu, D. "Radio Number for Trees." Disc. Math. 308, 1153-1164, 2008.

Liu, D. D.-F.; Zhu, X. "Multilevel Distance Labelings for Paths and Cycles." SIAM J. Disc. Math. 19, 610-621, 2005.

Zhang, P. "Radio Labeling of Cycles." Ars Combin. 65, 21-32, 2002.