Radio k-colorings of paths

Abstract

For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that d(u, v) + |c(u)− c(v)| ≥ 1 + k for every two distinct vertices u and v of G, where d(u, v) is the distance between u and v. The value rck(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rck(G) of G is the minimum value of rck(c) taken over all radio k-colorings c of G. In this paper, radio k-colorings of paths are studied. For the path Pn of order n ≥ 9 and n odd, a new improved bound for rcn−2(Pn) is presented. For n ≥ 4, it is shown that rcn−3(Pn) ≤ ∗Research supported in part by the Western Michigan University Arts and Sciences Teaching and Research Award Program. 6 G. Chartrand, L. Nebeský and P. Zhang ( n−2 2 ) + 2. Upper and lower bounds are also presented for rck(Pn) in terms of k when 1 ≤ k ≤ n−1. The upper bound is shown to be sharp when 1 ≤ k ≤ 4 and n is sufficiently large.