Geodesic
Nowhere-zero modular edge-graceful graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 487-505.

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For a connected graph G of order n ≥ 3, let f: E(G) → ℤₙ be an edge labeling of G. The vertex labeling f': V(G) → ℤₙ induced by f is defined as f'(u) = ∑_v ∈ N(u) f(uv), where the sum is computed in ℤₙ. 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) ≠ 0 for all e ∈ 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 ≥ 3 is nowhere-zero modular edge-graceful if and only if n ≢ 2 mod 4, G ≠ K₃ and G is not a star of even order. For a connected graph G of order n ≥ 3, the smallest integer k ≥ n for which there exists an edge labeling f: E(G) → ℤₖ - 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.
Keywords: modular edge-graceful labelings and graphs, nowhere-zero labelings, modular edge-gracefulness 
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Jones, Ryan; Zhang, Ping. Nowhere-zero modular edge-graceful graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 487-505. http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a8/

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