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Super edge-graceful paths and cycles. (English) Zbl 1214.05149
Summary: A graph \(G(V,E)\) of order \(|V|= p\) and size \(|E|= q\) is called super edge-graceful if there is a bijection \(f\) from \(E\) to \(\{0,\pm 1,\pm 2,\dots,\pm{q-1\over 2}\}\) when \(q\) is odd and from \(E\) to \(\{\pm 1,\pm 2,\dots,\pm{q\over 2}\}\) when \(q\) is even such that the induced vertex labeling \(f^*\) defined by \(f^*(x)= \sum_{xy\in E(G)} f(xy)\) over all edges \(xy\) is a bijection from \(V\) to \(\{0,\pm 1,\pm 2,\dots, \pm{p-1\over 2}\}\) when \(p\) is odd and from \(V\) to \(\{\pm 1,\pm 2,\dots, \pm{p\over 2}\}\) when \(p\) is even.
We prove that all paths \(P_n\), except \(P_2\) and \(P_4\) and all cycles except \(C_4\) and \(C_6\) are super edge-graceful.
MSC:
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C38 Paths and cycles
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Full Text: arXiv