Involve: A Journal of Mathematics

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

Prime labelings of infinite graphs

Matthew Kenigsberg and Oscar Levin

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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

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

Received: 22 February 2018
Revised: 9 July 2018
Accepted: 8 November 2018
First available in Project Euclid: 30 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

graph labelings infinite graphs prime labelings computability theory


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

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