Radio colorings of graphs – A survey

For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a k-radio coloring of G is an assignment c of colors (positive integers) to the vertices of G such that for every two distinct vertices u and v of G, where d(u, v) is the distance between u and v. The value of a k-radio coloring c of G is the maximum color assigned by c to a vertex of G. The k-radio coloring number of G is the minimum value of taken over all k-radio colorings c of G. We survey several results from this area of research including the related area of Hamiltonian colorings.