#### Vol. 9, No. 2, 2016

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Radio number for fourth power paths

### Min-Lin Lo and Linda Victoria Alegria

Vol. 9 (2016), No. 2, 317–332
##### 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 $\left\{0,1,2,\dots \phantom{\rule{0.3em}{0ex}}\right\}$ 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 $1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}8\right)$.

##### Keywords
channel assignment problem, multilevel distance labeling, radio number, radio labeling
Primary: 05C78
##### Milestones
Received: 24 November 2014
Revised: 12 April 2015
Accepted: 12 April 2015
Published: 2 March 2016

Communicated by Jerrold Griggs
##### Authors
 Min-Lin Lo Department of Mathematics California State University, San Bernardino San Bernardino, CA 92407 United States Linda Victoria Alegria Department of Mathematics California State University, San Bernardino San Bernardino, CA 92407 United States