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) = ∑ 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 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.