#### 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
##### 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 $\left\{1,2,\dots ,|V|\right\}$ 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
Primary: 11B39
Secondary: 05C05