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It is conjectured that trees are all harmonious if one vertex label is allowed to be reused. The seven-page book graph provides an example of a graph that is not harmonious.

Graph coloring

A graph coloring is a subclass of graph labelings. Vertex colorings assign different labels to adjacent vertices, while edge colorings assign different labels to adjacent edges.

Lucky labeling

A lucky labeling of a graph is an assignment of positive integers to the vertices

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