Vol. 14, No. 4, 2021

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On prime labelings of Fibonacci trees

Bayley Larsen, Hunter Lehmann, Andrew Park and Leanne Robertson

Vol. 14 (2021), No. 4, 595–603

A tree has a prime labeling and is said to be prime if there exists a bijection from its vertex set V to the set of integers {1,2,,|V |} 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 fn) 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 f32 1 = 2,178,308 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 f30 = 832,040 that begin with f29 = 514,229 and f31 = 1,346,269, respectively.

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prime labelings, coprime mappings, Fibonacci trees
Mathematical Subject Classification 2010
Primary: 11B39
Secondary: 05C05
Received: 28 December 2019
Revised: 27 February 2021
Accepted: 29 March 2021
Published: 23 October 2021

Communicated by Kenneth S. Berenhaut
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