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

Abstract

The radio k-chromatic number $r{c}_k\left(G\right)$ of a graph G is the minimum integer λ such that there exists a function $\varphi :V\left(G\right)\to \{0,1,\cdots ,\lambda \}$ satisfying $\left|\varphi \right(u)-\varphi (v\left)\right|\ge k+1-d(u,v)$, 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 $r{c}_k\left(G\right)$ 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 $r{c}_k\left(G\right)$ 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.