Natural CLIQUE to k-Color reduction
Given a graph $G$ and a number $k$, such that you want to know whether $G$ contains a $k$-clique, let n be the number of vertices in $G$. We construct another graph $H$, such that $H$ is $n$-colorable if and only if $G$ has a $k$-clique, as follows:

(1) For each vertex $v$ in $G$, make an $n$-clique of vertices $(v,i)$ in $H$, where $i$ ranges from $1$ to $n$.

(2) Add one additional vertex $x$ to $H$.

(3) For each triple $\{x,y,z\}$ of vertices in $H$, where $y=(v,i)$ and $z=(u,j)$, test whether one of the following conditions holds: either $u\ne v$ and $i=j$, or $u$ and $v$ are nonadjacent vertices in $G$ with $\max(i,j)\le k$. If either of these two things is true, add another $n$-clique to $H$. Within this clique, select three vertices $x'$, $y'$, and $z'$. Connect $x$ to every vertex in the clique except for $y'$ and $z'$; connect $y$ to every vertex in the clique except for $x'$ and $z'$; and connect $z$ to every vertex in the clique except for $x'$ and $y'$.

The gadgets added in step (3) prevent the triple of vertices $x$, $y$, and $z$ from all being given the same color as each other in a valid coloring of $H$. The clique in $G$ can be recovered from a coloring of $H$ as the set of vertices $(v,i)$ that are in the same color class as $x$ and that have $i\le k$.