#### 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 $\left\{1,2,\dots ,n\right\}$. 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
Primary: 05C78