# Nowhere-zero modular edge-graceful graphs

• Volume: 32, Issue: 3, page 487-505
• ISSN: 2083-5892

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## Abstract

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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 ${f}^{\text{'}}\left(u\right)={\sum }_{v\in N\left(u\right)}f\left(uv\right)$, 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.

## How to cite

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Ryan Jones, and Ping Zhang. "Nowhere-zero modular edge-graceful graphs." Discussiones Mathematicae Graph Theory 32.3 (2012): 487-505. <http://eudml.org/doc/271066>.

@article{RyanJones2012,
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 $f^\{\prime \}(u) = ∑_\{v ∈ N(u)\} f(uv)$, 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.},
author = {Ryan Jones, Ping Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {modular edge-graceful labelings and graphs; nowhere-zero labelings; modular edge-gracefulness},
language = {eng},
number = {3},
pages = {487-505},
title = {Nowhere-zero modular edge-graceful graphs},
url = {http://eudml.org/doc/271066},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Ryan Jones
AU - Ping Zhang
TI - Nowhere-zero modular edge-graceful graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 3
SP - 487
EP - 505
AB - 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 $f^{\prime }(u) = ∑_{v ∈ N(u)} f(uv)$, 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.
LA - eng
KW - modular edge-graceful labelings and graphs; nowhere-zero labelings; modular edge-gracefulness
UR - http://eudml.org/doc/271066
ER -

## References

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