Nowhere-zero modular edge-graceful graphs

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) = P 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) 6 0 for all e ∈ E(G) and in this case, G is a nowherezero 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 6≡2 (mod 4), G 6 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