Graceful Coloring of Ladder Graphs

A graceful k-coloring of a non-empty graph $G=(V,E)$ is a proper vertex coloring $f:V(G)\to \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},k\}$, $k\ge 2$, which induces a proper edge coloring $f^{\ast }:E(G)\to \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},k-1\}$ defined by $f^{\ast }(uv)=\mathrm{\mid }f(u)-f(v)\mathrm{\mid }$, where $u,v\in V(G)$. The minimum $k$ for which $G$ has a graceful $k$-coloring is called graceful chromatic number, ${\chi }_g(G)$. The graceful chromatic number for a few variants of ladder graphs are investigated in this article.