Vol. 12, No. 4, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 4, 541–720
Issue 3, 361–539
Issue 2, 181–360
Issue 1, 1–180

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Ethics Statement
Editorial Login
Author Index
Coming Soon
ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Other MSP Journals
Prime labelings of infinite graphs

Matthew Kenigsberg and Oscar Levin

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

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.

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

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