Edge-Graceful Graph

A (p,q)-graph is edge-graceful if the edges can be labeled 1 through q in such a way that the labels induced on the vertices by summing over incident edges modulo p are distinct. Lo (1985) showed that a graph G is edge-graceful only if p|q^2+q-p(p-1)/2. 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 C_3 with C_4 (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.

See also

Graceful Graph

Portions of this entry contributed by Adrian Riskin

Portions of this entry contributed by Georgia Weidman

Explore with Wolfram|Alpha


Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018., 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.

Referenced on Wolfram|Alpha

Edge-Graceful Graph

Cite this as:

Riskin, Adrian; Weidman, Georgia; and Weisstein, Eric W. "Edge-Graceful Graph." From MathWorld--A Wolfram Web Resource.

Subject classifications