Discussiones Mathematicae Graph Theory 32(3) (2012) 487-505
doi: 10.7151/dmgt.1621

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Nowhere-zero Modular Edge-graceful Graphs

Ryan Jones and Ping Zhang

Department of Mathematics
Western Michigan University
Kalamazoo, MI 49008, USA

Abstract

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

2010 Mathematics Subject Classification: 05C05, 05C15, 05C78.

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Received 1 June 2011
Revised 4 September 2011
Accepted 4 September 2011