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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
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Abstract |
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A graph has a prime labeling if its vertices can be assigned distinct numbers from 1 to 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 so that the 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 -partite -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. |
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Keywords
prime labeling, $k$-partite $k$-uniform hypergraph
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Mathematical Subject Classification
Primary: 05C78
Secondary: 05C65
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Milestones
Received: 1 September 2020
Revised: 1 October 2021
Accepted: 9 October 2021
Published: 29 July 2022
Communicated by Joseph Gallian
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Authors |
| Arran Hamm | |
| Department of Mathematics Winthrop University Rock Hill, SC United States |
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| Jessica Hamm | |
| Department of Mathematics Winthrop University Rock Hill, SC United States |
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| Alan Way | |
| Department of Mathematics Winthrop University Rock Hill, SC United States |
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