Vol. 14, No. 4, 2021

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Graceful wind chimes

Jocelyn R. Bell, Lotenna Nwobbi, Connor Parrow, Alexandria Wheeler and Ephrata Zelleke

Vol. 14 (2021), No. 4, 683–696
Abstract

A graceful labeling of a graph with n edges is an assignment to each vertex of a unique label between 0 and n, inclusive, satisfying the additional requirement that if each edge is labeled by the absolute value of the difference of its incident vertices, then the set of edge labels is exactly {1,2,,n}. It has been an open question for more than 50 years whether or not every tree has a graceful labeling. In fact, it is unknown whether every tree with maximum degree 3 has a graceful labeling. We introduce a class of trees with maximum degree 3, which we call wind chimes. We prove that all wind chimes satisfying certain criteria have a graceful labeling.

Keywords
graceful labeling, alpha labeling, graceful tree, wind chime
Mathematical Subject Classification
Primary: 05C78
Milestones
Received: 17 November 2020
Revised: 19 April 2021
Accepted: 19 April 2021
Published: 23 October 2021

Communicated by Joseph Gallian
Authors
Jocelyn R. Bell
Department of Mathematics and Computer Science
Hobart and William Smith Colleges
Geneva, NY
United States
Lotenna Nwobbi
Department of Mathematics
Harvey Mudd College
Claremont, CA
United States
Connor Parrow
Department of Mathematics and Computer Science
Hobart and William Smith Colleges
Geneva, NY
United States
Alexandria Wheeler
Department of Mathematics
Carthage College
Kenosha, WI
United States
Ephrata Zelleke
Center for Data, Mathematical & Computational Sciences
Goucher College
Baltimore, MD
United States