Further results on super graceful labeling of graphs

Let G = (V(G), E(G)) be a simple, finite and undirected graph of order p and size q. A bijection f : V(G). E(G) -> {k, k + 1, k + 2, ... , k + p + q - 1} such that f (uv) = vertical bar f(u) - f(v)vertical bar for every edge uv is an element of E(G) is said to be a k-super graceful labeling of G. We say G is k-super graceful if it admits a k-super graceful labeling. For k = 1, the function f is called a super graceful labeling and a graph is super graceful if it admits a super graceful labeling. In this paper, we study the super gracefulness of complete graph, the disjoint union of certain star graphs, the complete tripartite graphs K(1, 1, n), and certain families of trees. We also present four methods of constructing new super graceful graphs. In particular, all trees of order at most 7 are super graceful. We conjecture that all trees are super graceful. (C) 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license