is a labeling of the edges by distinct integers in such that the labeling on the vertices induced by labeling a vertex with the sum of the incident edges taken modulo assigns all values from 0 to to the vertices. A graph is said to be "edge-graceful" if it admits an edge-graceful labeling.

Edge-graceful labelings were first introduced by Sheng-Ping Lo in 1985.

A necessary condition for a graph to be edge-graceful is "Lo's condition":
:

Harmonious labeling

A "harmonious labeling" on a graph is an injection

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