Radio Number for Square Paths

Let G be a connected graph. For any two vertices u and v, let d(u, v) denote the distance between u and v in G. The maximum distance between any pair of vertices is called the diameter of G and denoted by diam(G). A radio-labeling (or multi-level distance labeling) with span k for G is a function f that assigns to each vertex with a label from the set {0, 1, 2, · · · , k} such that the following holds for any vertices u and v: |f(u) − f(v)| ≥ diam(G) − d(u, v) + 1. The radio number of G is the minimum span over all radio-labelings of G. The square of G is a graph constructed from G by adding edges between vertices of distance two apart in G. In this article, we completely determine the radio number for the square of any path.