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On the super edge-graceful trees of even orders. I. (English) Zbl 1119.05095

Let \(G\) be a graph with \(p\) vertices (called its order) and \(q\) edges (called its size). \(G\) is said to be edge-graceful if one can label its edges by \(1,2,\dots, q\) so that the vertex sums \(\pmod p\) are distinct. This paper is concerned with a special kind of labeling called super edge-graceful (SEG). A graph \(G= (V,E)\) where \(|V|= p\) and \(|E|= q\) is said to be SEG if there exists a bijection \(f: E\to\{0\pm 1,\pm 2,\dots,\pm{q-1\over 2}\}\) if \(q\) is odd and \(f: E\to\{\pm 1,\pm 2,\dots,\pm {q\over 2}\}\) if \(q\) is even, such that the induced vertex labeling \(f^*\) defined by \(f^*(u)= \{\Sigma f(u,v): (u,v)\in E\}\) having the property \(f^*: V\to\{0,\pm 1,\pm 2,\dots,\pm{p-1\over 2}\}\) if \(p\) is odd and \(f^*: V\to\{\pm 1,\dots, \pm{p\over 2}\}\) if \(p\) is even is a bijection.
The authors study the SEG labeling for some even trees. It is shown that trees of order 6, some trees of order 8 and of order 10 are SEG. The authors go on to discuss the SEG labelings of some spiders and some double stars.

MSC:

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C05 Trees

Keywords:

labeling
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