A linear algorithm for radio k-coloring of powers of paths having small diameters

Abstract

The radio k-chromatic number rck(G) of a graph G is the minimum integer λ such that there exists a function ϕ:V(G)→{0,1,⋯,λ} satisfying |ϕ(u)−ϕ(v)|≥k+1−d(u,v), where d(u,v) denotes the distance between u and v. A considerable amount of attention has been given to find the exact values or providing polynomial time algorithms to determine rck(G) for several basic graph families such as paths, cycles, trees, and powers of paths, usually for some specific values of k.In this article, we find the exact values of rck(G) where G is a power of a path with diameter strictly less than k. Our proof readily provides a linear time algorithm for assigning a radio k-coloring of G. Furthermore, our proof technique is a potential tool for solving the same problem for other classes of graphs having “small” diameters.