On $k$-Super Graceful Labeling of Graphs

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to {k, k+1, k+2, \ldots, k+p+q-1}$suchthat$f(uv)= |f(u) - f(v)|$foreveryedge$uv\in E(G)$issaidtobea$k$-supergracefullabelingof$G$\mathrm{.}Wesay$G$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.