## 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

 [1] G. Chartrand and L. Leśniak, Graphs and Digraphs (Wadsworth &Brooks/Cole Advanced Books and Software, Monterey, Calif. 1986). [2] J. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 5 (2002) #DS6. [3] R.L. Graham and N.J.A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Discrete Math. 1 (1980) 382-404, doi: 10.1137/0601045. [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. [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. [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] S.C. Shee, On harmonious and related graphs, Ars Combin. 23 (1987) (A) 237-247. [8] S.C. Shee, Some results on λ-valuation of graphs involving complete bipartite graphs, Discrete Math. 28 (1991) 1-14.