On k-Super Graceful Labeling of Graphs: Gee-Choon Lau, Wai-Chee Shiu, Ho-Kuen Ng - Team, Support (Thai Journal of Mathematics, 2022)
Let G = (V(G), E(G)) be a simple, finite and undirected graph of order p and size q. For k >= 1, a bijection f: V(G) U E(G) -> k, k+1, ..., k+p+q-1 such that f(uv) = |f(u) - f(v)| for every edge uv in 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. In this paper, we study the k-super gracefulness of some standard graphs. Some general properties are obtained. Particularly, we found many sufficient conditions on k-super gracefulness for many families of (complete) bipartite and tripartite graphs. We show that some of the conditions are also necessary.