Graham--Sloane harmonious tree conjecture
Graham--Sloane harmonious tree conjecture
Let be a tree with vertices. A harmonious labelling assigns distinct vertex labels and gives an edge the label . Harmonious Tree Conjecture. For any -vertex tree there exists an injective labelling
such that the values
are pairwise distinct. This is the harmonious analogue of the graceful tree conjecture, introduced by Graham and Sloane; the source gives no resolution and the problem remains open.
Sources & referencesView supporting material
Primary source
Anna Adamaszek, Peter Allen, Codrut Grosu and Jan Hladky, “Almost all trees are almost graceful”, arXiv:1608.01577 (2019).
Progress summary
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