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, vertical bar f(u) -f(v)vertical bar >= 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 rn(k)(G), is the smallest span among all radio-k-labelings admitted by G. A cycle C-n has diameter d = Left perpendicularn/2right perpendicular. In this paper, we combine a lower bound approach with cyclic group structure to determine the value of rn(k)(C-n) for k >= n - 3. For d <= k < n - 3, we obtain the values of rn(k)(C-n) 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 rn(d)(C-n) and rn(d+1)(C-n) shown by Liu and Zhu (2005), and by Karst et al. (2017), respectively. (C) 2022 The Author(s). Published by Elsevier B.V.
Frequency assignment problemRadio labelingRadio-k-labelingRadio-k-numberCyclic groupsNUMBERASSIGNMENTGRAPHS
NSTL
Elsevier
