Labeling the vertex amalgamation of graphs

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. 130 R.M. Figueroa-Centeno, R. Ichishima, F.A. Muntaner-Batle 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.