Radio number for fourth power paths

Abstract

Let $G$ be a connected graph. For any two vertices $u$ and $v$, let $d\left(u,v\right)$ denote the distance between $u$ and $v$ in $G$. The maximum distance between any pair of vertices of $G$ is called the diameter of $G$ and denoted by $diam\left(G\right)$. A radio labeling (or multilevel distance labeling) of $G$ is a function $f$ that assigns to each vertex a label from the set $\left\{0,1,2,\dots \right\}$ such that the following holds for any vertices $u$ and $v$: $|f\left(u\right)-f\left(v\right)|\ge diam\left(G\right)-d\left(u,v\right)+1$. The span of $f$ is defined as $\underset{u,v\in V\left(G\right)}{max}\left\{|f\left(u\right)-f\left(v\right)|\right\}$. The radio number of $G$ is the minimum span over all radio labelings of $G$. The fourth power of $G$ is a graph constructed from $G$ by adding edges between vertices of distance four or less apart in $G$. In this paper, we completely determine the radio number for the fourth power of any path, except when its order is congruent to $1\left(mod8\right)$.