A graceful labeling of a graph with
edges is an assignment to each vertex of a unique label between
and
,
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
.
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.
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