## Involve: A Journal of Mathematics

- Involve
- Volume 9, Number 2 (2016), 317-332.

### Radio number for fourth power paths

Min-Lin Lo and Linda Victoria Alegria

#### 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 $1\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}8\right)$.

#### Article information

**Source**

Involve, Volume 9, Number 2 (2016), 317-332.

**Dates**

Received: 24 November 2014

Revised: 12 April 2015

Accepted: 12 April 2015

First available in Project Euclid: 22 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1511371003

**Digital Object Identifier**

doi:10.2140/involve.2016.9.317

**Mathematical Reviews number (MathSciNet)**

MR3470734

**Zentralblatt MATH identifier**

1333.05266

**Subjects**

Primary: 05C78: Graph labelling (graceful graphs, bandwidth, etc.)

**Keywords**

channel assignment problem multilevel distance labeling radio number radio labeling

#### Citation

Lo, Min-Lin; Alegria, Linda Victoria. Radio number for fourth power paths. Involve 9 (2016), no. 2, 317--332. doi:10.2140/involve.2016.9.317. https://projecteuclid.org/euclid.involve/1511371003