# Labeling the vertex amalgamation of graphs

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

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## Abstract

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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 $f:V\left(G\right)\to {Z}_{q}$ 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 $f:V\left(G\right)\to {Z}_{q+1}$ 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.

## How to cite

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Ramon 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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4. [4] K.M. Koh, D.G. Rogers, P.Y. Lee and C.W. Toh, On graceful graphs V: unions of graphs with one vertex in common, Nanta Math. 12 (1979) 133-136. Zbl0428.05048
5. [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
6. [6] A. Rosa, On certain valuations of the vertices of a graph, in: Theory of Graphs (Internat. Symposium, Rome, July 1966, Gordon and Breach, N.Y. and Dunod Paris, 1967) 87-95.
7. [7] S.C. Shee, On harmonious and related graphs, Ars Combin. 23 (1987) (A) 237-247. Zbl0616.05055
8. [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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