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Nowhere-zero modular edge-graceful graphs. (English) Zbl 1257.05151

Summary: For a connected graph \(G\) of order \(n \geq 3\), let \(f: E(G) \to {\mathbb{Z}}_n\) be an edge labeling of \(G\). The vertex labeling \(f' : V(G) \to {\mathbb{Z}}_n\) induced by \(f\) is defined as \(f'(u) = \sum_{v \in N(u)} f(uv)\), where the sum is computed in \({\mathbb{Z}}_n\). If \(f'\) is one-to-one, then \(f\) is called a modular edge-graceful labeling and \(G\) is a modular edge-graceful graph. A modular edge-graceful labeling \(f\) of \(G\) is nowhere-zero if \(f(e) \neq 0\) for all \(e \in E(G)\) and in this case, \(G\) is a nowhere-zero modular edge-graceful graph.
It is shown that a connected graph \(G\) of order \(n \geq 3\) is nowhere-zero modular edge-graceful if and only if \(n \not\equiv 2 \pmod 4\), \(G \neq K_3\) and \(G\) is not a star of even order. For a connected graph \(G\) of order \(n \geq 3\), the smallest integer \(k \geq n\) for which there exists an edge labeling \(f : E(G) \to {\mathbb{Z}}_{k}-\{0\}\) such that the induced vertex labeling \(f'\) is one-to-one is referred to as the nowhere-zero modular edge-gracefulness of \(G\) and this number is determined for every connected graph of order at least 3.

MSC:

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C05 Trees
05C15 Coloring of graphs and hypergraphs
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