# Labeling the vertex amalgamation of graphs

Ramon M. Figueroa-Centeno; Rikio Ichishima; Francesc A. Muntaner-Batle

Discussiones Mathematicae Graph Theory (2003)

- Volume: 23, Issue: 1, page 129-139
- ISSN: 2083-5892

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topRamon M. Figueroa-Centeno, Rikio Ichishima, and Francesc A. Muntaner-Batle. "Labeling the vertex amalgamation of graphs." Discussiones Mathematicae Graph Theory 23.1 (2003): 129-139. <http://eudml.org/doc/270335>.

@article{RamonM2003,

abstract = {A graph G of size q is graceful if there exists an injective function f:V(G)→ 0,1,...,q such that each edge uv of G is labeled |f(u)-f(v)| and the resulting edge labels are distinct. Also, a (p,q) graph G with q ≥ p is harmonious if there exists an injective function $f:V(G) → Z_q$ such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct, whereas G is felicitous if there exists an injective function $f: V(G) → Z_\{q+1\}$ such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct. In this paper, we present several results involving the vertex amalgamation of graceful, felicitous and harmonious graphs. Further, we partially solve an open problem of Lee et al., that is, for which m and n the vertex amalgamation of n copies of the cycle Cₘ at a fixed vertex v ∈ V(Cₘ), Amal(Cₘ,v,n), is felicitous? Moreover, we provide some progress towards solving the conjecture of Koh et al., which states that the graph Amal(Cₘ,v,n) is graceful if and only if mn ≡ 0 or 3 mod 4. Finally, we propose two conjectures.},

author = {Ramon M. Figueroa-Centeno, Rikio Ichishima, Francesc A. Muntaner-Batle},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {felicitous labellings; graceful labellings; harmonious labellings.; graph labeling; graceful; harmonious; felicitous; vertex amalgamation},

language = {eng},

number = {1},

pages = {129-139},

title = {Labeling the vertex amalgamation of graphs},

url = {http://eudml.org/doc/270335},

volume = {23},

year = {2003},

}

TY - JOUR

AU - Ramon M. Figueroa-Centeno

AU - Rikio Ichishima

AU - Francesc A. Muntaner-Batle

TI - Labeling the vertex amalgamation of graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2003

VL - 23

IS - 1

SP - 129

EP - 139

AB - A graph G of size q is graceful if there exists an injective function f:V(G)→ 0,1,...,q such that each edge uv of G is labeled |f(u)-f(v)| and the resulting edge labels are distinct. Also, a (p,q) graph G with q ≥ p is harmonious if there exists an injective function $f:V(G) → Z_q$ such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct, whereas G is felicitous if there exists an injective function $f: V(G) → Z_{q+1}$ such that each edge uv of G is labeled f(u) + f(v) mod q and the resulting edge labels are distinct. In this paper, we present several results involving the vertex amalgamation of graceful, felicitous and harmonious graphs. Further, we partially solve an open problem of Lee et al., that is, for which m and n the vertex amalgamation of n copies of the cycle Cₘ at a fixed vertex v ∈ V(Cₘ), Amal(Cₘ,v,n), is felicitous? Moreover, we provide some progress towards solving the conjecture of Koh et al., which states that the graph Amal(Cₘ,v,n) is graceful if and only if mn ≡ 0 or 3 mod 4. Finally, we propose two conjectures.

LA - eng

KW - felicitous labellings; graceful labellings; harmonious labellings.; graph labeling; graceful; harmonious; felicitous; vertex amalgamation

UR - http://eudml.org/doc/270335

ER -

## References

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- [5] S.M. Lee, E. Schmeichel and S.C. Shee, On felicitous graphs, Discrete Math. 93 (1991) 201-209, doi: 10.1016/0012-365X(91)90256-2. Zbl0741.05059
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- [7] S.C. Shee, On harmonious and related graphs, Ars Combin. 23 (1987) (A) 237-247. Zbl0616.05055
- [8] S.C. Shee, Some results on λ-valuation of graphs involving complete bipartite graphs, Discrete Math. 28 (1991) 1-14.

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