## Involve: A Journal of Mathematics

- Involve
- Volume 12, Number 4 (2019), 633-646.

### Prime labelings of infinite graphs

Matthew Kenigsberg and Oscar Levin

#### Abstract

A finite graph on $n$ vertices has a prime labeling provided there is a way to label the vertices with the integers 1 through $n$ such that every pair of adjacent vertices has relatively prime labels. We extend the definition of prime labeling to infinite graphs and give a simple necessary and sufficient condition for an infinite graph to have a prime labeling. We then measure the complexity of prime labelings of infinite graphs using techniques from computability theory to verify that our condition is as simple as possible.

#### Article information

**Source**

Involve, Volume 12, Number 4 (2019), 633-646.

**Dates**

Received: 22 February 2018

Revised: 9 July 2018

Accepted: 8 November 2018

First available in Project Euclid: 30 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1559181656

**Digital Object Identifier**

doi:10.2140/involve.2019.12.633

**Mathematical Reviews number (MathSciNet)**

MR3941602

**Zentralblatt MATH identifier**

07072543

**Subjects**

Primary: 05C78: Graph labelling (graceful graphs, bandwidth, etc.) 05C63: Infinite graphs 05C85: Graph algorithms [See also 68R10, 68W05] 03D80: Applications of computability and recursion theory

**Keywords**

graph labelings infinite graphs prime labelings computability theory

#### Citation

Kenigsberg, Matthew; Levin, Oscar. Prime labelings of infinite graphs. Involve 12 (2019), no. 4, 633--646. doi:10.2140/involve.2019.12.633. https://projecteuclid.org/euclid.involve/1559181656