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On edge-graceful labelings of graphs. (English) Zbl 0597.05054
Proc. Conf., Sundance/Utah 1985, Congr. Numerantium 50, 231-241 (1985).

[For the entire collection see Zbl 0583.00003.]

This paper is a further one dealing with labeling graphs. The author considers edge-graceful graphs being defined as follows: A connected graph $G=\left(V\left(G\right),E\left(G\right)\right)$ is said to be edge-graceful if, and only if, there is an edge-labeling $g:E\to \left\{1,2,···,|\left(G\right)|\right\}$ of $G$ such that the weights

${w}_{g}\left(v\right)=\sum _{e\in E\left(G\right),\phantom{\rule{4pt}{0ex}}e\phantom{\rule{4.pt}{0ex}}\text{incident}\phantom{\rule{4.pt}{0ex}}\text{to}\phantom{\rule{4.pt}{0ex}}v}g\left(e\right)\phantom{\rule{1.em}{0ex}}\left(\text{mod}|V\left(G\right)|\right)$

for each vertex $v\in V\left(G\right)$ are consecutive integers ranging from 0 to $|V\left(G\right)|-1$. Analogously to the theory of the graceful graphs the author derives some properties of edge- graceful graphs, proves a necessary condition for a graph to be edge- graceful, determines some classes of edge-graceful graphs, and finishes with the conjecture: Each complete graph ${K}_{p}$ $\left(p\ge 2\right)$ is edge-graceful.

Reviewer: R. Bodendiek

##### MSC:
 05C99 Graph theory