A Linear Algorithm for Radio k-Coloring Powers of Paths Having Small Diameter

The radio k-chromatic number $$rc_k(G)$$ of a graph G is the minimum integer $$\lambda $$ such that there exists a function $$\phi : V(G) \rightarrow \{0,1,\cdots , \lambda \}$$ satisfying $$|\phi (u)-\phi (v)| \ge k+1 - d(u,v)$$ , where d(u, v) denotes the distance between u and v. To date, several upper and lower bounds of $$rc_k(\cdot )$$ is established for different graph families. One of the most notable works in this domain is due to Liu and Zhu [SIAM Journal on Discrete Mathematics 2005] whose main results were computing the exact values of $$rc_k(\cdot )$$ for paths and cycles for the specific case when k is equal to the diameter. In this article, we find the exact values of $$rc_k(G)$$ for powers of paths where the diameter of the graph is strictly less than k. Our proof readily provides a linear time algorithm for providing such labeling. Furthermore, our proof technique is a potential tool for solving the same problem for other graph classes with “small” diameter.