Vol. 12, No. 4, 2019

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Prime labelings of infinite graphs

Matthew Kenigsberg and Oscar Levin

Vol. 12 (2019), No. 4, 633–646
DOI: 10.2140/involve.2019.12.633
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.

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Keywords
graph labelings, infinite graphs, prime labelings, computability theory
Mathematical Subject Classification 2010
Primary: 05C78, 05C63, 05C85, 03D80
Milestones
Received: 22 February 2018
Revised: 9 July 2018
Accepted: 8 November 2018
Published: 16 April 2019

Communicated by Kenneth S. Berenhaut
Authors
Matthew Kenigsberg
Vanderbilt University
Nashville, TN
United States
Oscar Levin
School of Mathematical Sciences
University of Northern Colorado
Greeley, CO
United States