NOWHERE-ZERO MODULAR EDGE-GRACEFUL GRAPHS

For a connected graph G of order n >= 3, let f : E(G) -> Z(n) be an edge labeling of G. The vertex labeling f' : V(G) -> Z(n) induced by f is defined as f'(u) Sigma(v is an element of N(u)) f (uv), where the sum is computed in 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) not equal 0 for all e is an element of 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 not equivalent to 2 (mod 4), G not equal k(3) (a)nd 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) -> 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.