Further results on super graceful labeling of graphs

Let G=(V(G),E(G))G=(V(G),E(G)) be a simple, finite and undirected graph of order pp and size qq. A bijection f:V(G)∪E(G)→{k,k+1,k+2,…,k+p+q−1}f:V(G)∪E(G)→{k,k+1,k+2,…,k+p+q−1} such that f(uv)=|f(u)−f(v)|f(uv)=|f(u)−f(v)| for every edge uv∈E(G)uv∈E(G) is said to be a kk-super graceful labeling of GG. We say GG is kk-super graceful if it admits a kk-super graceful labeling. For k=1k=1, the function ff 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)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.