Edge-Graceful Graph
المؤلف:
Gallian, J.
المصدر:
"Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.
الجزء والصفحة:
...
6-5-2022
1704
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.
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