|
|||||
 |
|
|
|
|
|
|
|
On prime
labelings of Fibonacci trees
Bayley Larsen, Hunter Lehmann, Andrew Park and Leanne Robertson
Vol. 14 (2021), No. 4, 595–603
|
Abstract |
|
A tree has a prime labeling and is said to be prime if there exists a bijection from its vertex set to the set of integers such that adjacent vertices have coprime labels. Around 1980, Entringer and Tout conjectured that all trees have a prime labeling, but the conjecture remains open today. We study Fibonacci trees and prove that a special case (involving the Fibonacci numbers ) of a conjecture about coprime mappings implies that all Fibonacci trees are prime. We propose an algorithm for constructing the needed coprime mappings and use it to show that the first 30 Fibonacci trees are prime, the largest of which has vertices. This computation also supports the conjecture about coprime mappings by providing large examples of a coprime mapping on adjacent sets, the largest of which is on the sets of cardinality that begin with and , respectively. |
Keywords
prime labelings, coprime mappings, Fibonacci trees
|
Mathematical Subject Classification 2010
Primary: 11B39
Secondary: 05C05
|
Milestones
Received: 28 December 2019
Revised: 27 February 2021
Accepted: 29 March 2021
Published: 23 October 2021
Communicated by Kenneth S. Berenhaut
|
Authors |
| Bayley Larsen | |
| Department of Mathematics Seattle University Seattle, WA United States |
|
| Hunter Lehmann | |
| Department of Mathematics Seattle University Seattle, WA United States |
|
| Andrew Park | |
| Department of Mathematics Seattle University Seattle, WA United States |
|
| Leanne Robertson | |
| Department of Mathematics Seattle University Seattle, WA United States |
|
