A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma a...

A graph labeling is an assignment of integers to the vertices (or) edges or both subject to certain conditions. Graceful labeling was introduced by Rosa [6] while the concept of odd graceful labeling was introduced by Gnanajothi [2]. A Graph G (V, E), |V(G)| = p, |E(G)| = q is said to be odd graceful if there is an injection from V(G) to {0, 1, 2, .... 2q-1} such that when each edge xy is assig...

Let G(V,E) be a graph of order n and size m. A graceful labeling of G is an injection f : V (G) → {0, 1, 2, ...,m} such that, when each edge uv is assigned the label f(uv) = |f(u)− f(v)|, the resultant edge labels are distinct. We focus on general results in graceful labeling, and provide an affirmative answer to the following open problem: Can every connected graph be embedded as an induced su...