On prime labelings of Fibonacci trees

Larsen, Bayley; Lehmann, Hunter; Park, Andrew; Robertson, Leanne

doi:10.2140/involve.2021.14.595

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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

DOI: 10.2140/involve.2021.14.595

Abstract

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 ${1, 2, \dots, | V |}$ 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 $f_{n}$ ) 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 $f_{32} - 1 = 2, 178, 308$ 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 $f_{30} = 832, 040$ that begin with $f_{29} = 514, 229$ and $f_{31} = 1, 346, 269$ , respectively.

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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

Vol. 14, No. 4, 2021