A -graph
is edge-graceful if the edges can be labeled 1 through
in such a way that the labels induced on the vertices by summing over incident edges
modulo
are distinct. Lo (1985) showed that a graph is edge-graceful only if . Since then, many families of graphs have been
shown to be edge-graceful. These are exhaustively enumerated in Gallian's dynamic
survey, which also contains a complete bibliography of the subject.

In 1964, Ringel and Kotzig conjectured that every tree of odd order is edge-graceful. No known connected graph which satisfies Lo's condition has failed to be edge-graceful.
The simplest known graph which satisfies the condition and yet fails to be edge-graceful
is the disjoint union of with (Lee et al. 1992). A later proof by Riskin and Wilson
(1998) constructs infinite families of disjoint unions of cycles which satisfy Lo's
condition and yet fail to be edge-graceful.

Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin.DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Lee,
S. M., Lo, S. P., and Seah, E. "On Edge-Gracefulness of 2-Regular
Graphs." J. Combin. Math. Combin. Comput.12, 109-117, 1992.Lo,
S. P. "On Edge Graceful Labelings of Graphs." Congr. Numer.50,
231-241, 1985.Riskin, A. and Wilson, S. "Edge Graceful Labelings
of Disjoint Unions of Cycles." Bull. I.C.A.22, 53-58, 1998.Sheng-Ping,
L. "One Edge-Graceful Labeling of Graphs." Congr. Numer.50,
31-241, 1985.