The Radio Number of Gear Graphs

Let d(u, v) denote the distance between two distinct vertices of a connected graph G, and diam(G) be the diameter of G. A radio labeling c of G is an assignment of positive integers to the vertices of G satisfying d(u, v) + |c(u) − c(v)| ≥ diam(G) + 1. The maximum integer in the range of the labeling is its span. The radio number of G, rn(G), is the minimum possible span. The family of gear graphs of order n, Gn, consists of planar graphs with 2n + 1 vertices and 3n edges. We prove that the radio number of the n-gear is 4n + 2. 2000 AMS Subject Classification: 05C78 (05C15)