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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