Discussiones Mathematicae Graph Theory 23(1) (2003) 129-139
doi: 10.7151/dmgt.1190

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LABELING THE VERTEX AMALGAMATION OF GRAPHS

Ramon M. Figueroa-Centeno

 Mathematics Department
University of Hawaii-Hilo
200 W. Kawili St. Hilo, HI 96720, USA
e-mail: ramonf@hawaii.edu

Rikio Ichishima

College of Humanities and Sciences, Nihon University
3-25-40 Sakurajosui Setagaya-ku, Tokyo 156-8550, Japan
e-mail: ichishim@chs.nihon-u.ac.jp

Francesc A. Muntaner-Batle

Department de Matemàtica i Telemàtica
Universitat Politècnica de Catulunya
08071 Barcelona, Spain
e-mail: muntaner@mat.upc.es

Dedicated to Professor Miguel Angel Fiol

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)→Zq 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)→ Zq+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.

 Keywords: felicitous labellings, graceful labellings, harmonious labellings.

 2000 Mathematics Subject Classification: 05C78.

References

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Received 18 July 2001