Improved bounds on the Radio degree of a cycle

Abstract

Abstract A labeling f : V (G) → Z + such that |f(u)−f(v)| ≥ diam(G)+1−d(u, v) holds for every pair of vertices, u, v ∈ V (G), is called a radio labeling of a graph, G. The radio degree of a labeling, f : V (G) → {1, 2,… |V (G)|} in a graph, was defined by the same authors as the number of pairs of vertices u, v ∈ V (G) satisfying the condition |f(u) − f(v)| ≥ diam(G) + 1 − d(u, v) and was denoted by rdeg(f). The maximum value of rdeg(f) taken over all such labelings was defined as the radio degree of the graph, denoted by rdeg(G). The radio degree of some standard graphs like paths, complete graphs, complete bipartite graphs, wheel graph and fan graph was completely determined and a lower bound on the radio degree of cycles was obtained. In this paper, the authors have obtained better bounds on the radio degree of a cycle. AMS Subject Classification number: 05C78, 05C12, 05C15