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 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 distance between u and v. To date, several upper lower bounds $$rc_k(\cdot )$$ established for different families. One most notable works in this domain due to Liu Zhu [SIAM Journal on Discrete Mathematics 2005] whose main results were computing exact values paths cycles specific case when k equal diameter. In article, we find powers diameter strictly less than k. Our proof readily provides linear time algorithm providing labeling. Furthermore, our technique potential tool solving same problem other classes with “small”