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We define an $n$-gon to be any convex polygon with $n$ vertices. Let $V$represent the set of vertices of the polygon. A proper $k$-coloring refers to afunction, $f$ : $V$ $\rightarrow$ $\{1, 2, 3, ... k\}$, such that for any twovertices $u$ and $v$, if $f(u) = f(v)$, $u$ is not adjacent to $v$. The purposeof this paper is to develop a recursive algorithm to compute the number ofproper $k$-colorings in an $n$-gon. The proposed algorithm can easily be solvedto obtain the explicit expression. This matches the explicit expressionobtained from the popular conventional solutions, which confirms thecorrectness of the proposed algorithm. Often, for huge values of $n$ and $k$,it becomes impractical to display the output numbers, which would consist ofthousands of digits. We report the answer modulo a certain number. In suchsituations, the proposed algorithm is observed to run slightly faster than theconventional recursive algorithm.

On the number of proper $k$-colorings in an $n$-gon