Edge-Graceful Graph

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.

REFERENCES

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.