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

Abstract

The radio $k$-chromatic number $rc_k(G)$ of a graph $G$ is the minimum integer $λ$ such that there exists a function $φ: V(G) \to \{0,1,\cdots, λ\}$ satisfying $|φ(u)-φ(v)| \geq k+1 - d(u,v)$, 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 $rc_k(G)$ 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 $rc_k(G)$ 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.

Figures (4)

• Figure 1: Case 1.$n=16$, $m=4$, $diam(P_{16}^4)=4$, $k=6$.
• Figure 2: Case 2.$n=20$, $m=4$, $diam(P_{20}^4)=5$, $k=7$.
• Figure 3: Case 3.$n=19$, $m=4$, $diam(P_{19}^4)=5$, $k=7$, $s=3$.
• Figure 4: Case 4.$n=14$, $m=4$, $diam(P_{14}^4)=4$, $k=6$, $s=2$.

Theorems & Definitions (20)

• Remark \oldthetheorem
• Remark \oldthetheorem
• Theorem \oldthetheorem
• Theorem \oldthetheorem
• proof
• Lemma \oldthetheorem
• proof
• Lemma \oldthetheorem
• proof
• Lemma \oldthetheorem