On radio k-labeling of the power of the infinite path

Abstract

The radio k-chromatic number rck(G) of a graph G is the minimum integer ℓ such that there is a mapping f from the vertices of G to the set of integers {0,1,…,ℓ} satisfying |f(u)−f(v)|≥k+1−d(u,v) for any two distinct vertices u,v∈V(G), where d(u,v) denotes the distance between u and v. To date, the radio k-chromatic number of finite paths and square of finite paths is computed exactly when k is equal to the diameter of the graph. Moreover, the exact value of the radio k-chromatic number of an arbitrary power of a finite path is also computed when the diameter of the path is strictly smaller than k. Close lower and upper bounds for the radio k-chromatic number are given for the infinite path. Furthermore, lower and upper bounds have been found for an arbitrary power greater than or equal to two of infinite path. In this article, we improve the lower bound for rck(G) of any arbitrary power of the infinite path for any k. In particular, when k is even, we halve the gap between the known lower and upper bounds.