Prime labelings on a 3xn grid graph

Authors

Stephen J. Curran, University of Pittsburgh - JohnstownFollow
Matt A. Ollis, Emerson CollegeFollow

Abstract

It is conjectured that the mxn grid graph has a prime labeling for all positive integers m and n. It is known that for any prime p and any integer n such that 1≤n≤p², there exists a prime labeling on the pxn grid graph P_m x P_n. Also, it is known that the ladder P₂ x P_n has a prime labeling for all positive integers n. We assume that Goldbach's Even Conjecture and a strengthened variant of Lemoine's Conjecture are true in order to show that the 3xn grid graph P₃ x P_n has a prime labeling for every positive integer n. As a result, P₃ x P_n has a prime labeling for every positive integer n≤10⁷.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Curran, Stephen J. and Ollis, Matt A. (2025) "Prime labelings on a 3xn grid graph," Theory and Applications of Graphs: Vol. 12: Iss. 1, Article 4.
DOI: 10.20429/tag.2025.120104
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/4