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.

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