Radio-[formula omitted]-labeling of cycles for large [formula omitted

Abstract

Let G be a simple connected graph. For any two vertices u and v, let d(u,v) denote the distance between u and v in G. A radio-k-labeling of G for a fixed positive integer k is a function f which assigns to each vertex a non-negative integer label such that for every two vertices u and v in G, |f(u)−f(v)|⩾k−d(u,v)+1. The span of f is the difference between the largest and smallest labels of f(V). The radio-k-number of a graph G, denoted by rnk(G), is the smallest span among all radio-k-labelings admitted by G. A cycle Cn has diameter d=⌊n/2⌋. In this paper, we combine a lower bound approach with cyclic group structure to determine the value of rnk(Cn) for k⩾n−3. For d⩽k<n−3, we obtain the values of rnk(Cn) when n and k have the same parity, and prove partial results when n and k have different parities. Our results extend the known values of rnd(Cn) and rnd+1(Cn) shown by Liu and Zhu (2005), and by Karst et al. (2017), respectively.