  A graceful $l$-coloring of a graph $G$ is a proper vertex coloring with $l$
colors which induces a proper edge coloring with at most $l-1$ colors, where
the color for an edge $ab$ is the absolute difference between the colors
assigned to the vertices $a$ and $b$. The graceful chromatic number $\chi_g(G)$
is the smallest $l$ for which $G$ permits graceful $l$-coloring. The problem of
computing the graceful chromatic number of regular graphs is still open, though
the existence of the lower bound was proved in \cite{3}. Hence, we pay
attention to the computation of the graceful chromatic number of a special
class of regular graphs namely complete graphs using set theoretic approach.
Also, a few characterization of graphs based on their graceful chromatic number
were examined.
