Nowhere-zero modular edge-graceful graphs
Ryan Jones, Ping Zhang · Discussiones Mathematicae Graph Theory · 2012
For a connected graph G of order n ≥ 3, let f : E(G) → Z n be an edge labeling of G.The vertex labelingis 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 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 ≡ 2 (mod 4), G = K 3 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) → 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.