Vol. 15, No. 2, 2022

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A note on prime labeling $k$-partite $k$-graphs

Arran Hamm, Jessica Hamm and Alan Way

Vol. 15 (2022), No. 2, 233–239
Abstract

A graph has a prime labeling if its vertices can be assigned distinct numbers from 1 to |V | so that the vertices on each edge receive relatively prime labels. This definition can be extended naturally to hypergraphs, whose edges may contain more than two vertices, in the following way. A hypergraph has a prime labeling if its vertices can be assigned distinct numbers from 1 to |V | so that the gcd of numbers within each edge is 1 (which is sensible since greatest common divisor is defined for sets of numbers).

We examine the problem of prime labeling complete k-partite k-uniform hypergraphs. We prove that if this type of hypergraph has enough vertices and every pod of vertices is large enough, then it does not have a prime labeling. We also prove, on the other hand, that if a pod of vertices is small enough, then it does have a prime labeling.

Keywords
prime labeling, $k$-partite $k$-uniform hypergraph
Mathematical Subject Classification
Primary: 05C78
Secondary: 05C65
Milestones
Received: 1 September 2020
Revised: 1 October 2021
Accepted: 9 October 2021
Published: 29 July 2022

Communicated by Joseph Gallian
Authors
Arran Hamm
Department of Mathematics
Winthrop University
Rock Hill, SC
United States
Jessica Hamm
Department of Mathematics
Winthrop University
Rock Hill, SC
United States
Alan Way
Department of Mathematics
Winthrop University
Rock Hill, SC
United States