  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 $f: V(G)\cup E(G) \to \{k, k+1, k+2,
\ldots, k+p+q-1\}$ 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.
