Radio k-labeling of paths

The Channel Assignment Problem (CAP) is the problem of assigning channels (non-negative integers) to the transmitters in an optimal way such that interference is avoided. The problem, often modeled as a labeling problem on the graph where vertices represent transmitters and edges indicate closeness of the transmitters. A radio k-labeling of graphs is a variation of CAP. For a simple connected graph G = (V (G), E(G)) and a positive integer k with 1 ≤ k ≤ diam(G), a radio k-labeling of G is a mapping f: V (G) → {0, 1, 2, …} such that |f(u) − f(v)| ≥ k + 1 − d(u, v) for each pair of distinct vertices u and v of G, where diam(G) is the diameter of G and d(u, v) is the distance between u and v in G. The span of a radio k-labeling f is the largest integer assigned to a vertex of G. The radio k-chromatic number of G, denoted by rc_k (G), is the minimum of spans of all possible radio{⌈k-labelings⌉ of G. This} article presents the {⌈exact value⌉ of rc_k (P}_n) for even integer k ∈_2(n−2) 5, …, n − 2 and odd integer k ∈_2n+1 7, …, n − 1, i.e., at least 65% cases the radio k-chromatic number of the path P_n are obtain for fixed but arbitrary values of n. Also an improvement of existing lower bound of rc_k (P_n) has been presented for all values of k.