  Let $\mathcal{C}_4(n)$ be the family of all connected $4$-chromatic graphs of
order $n$. Given an integer $x\geq 4$, we consider the problem of finding the
maximum number of $x$-colorings of a graph in $\mathcal{C}_4(n)$. It was
conjectured that the maximum number of $x$-colorings is equal to
$(x)_{\downarrow 4}(x-1)^{n-4}$ and the extremal graphs are those which have
clique number $4$ and size $n+2$.
  In this article, we reduce this problem to a \textit{finite} family of
graphs. We show that there exist a finite family $\mathcal{F}$ of connected
$4$-chromatic graphs such that if the number of $x$-colorings of every graph
$G$ in $\mathcal{F}$ is less than $(x)_{\downarrow 4}(x-1)^{|V(G)|-4}$ then the
conjecture holds to be true.
