Let $n$ and $k$ be positive integers. A simple graph $G$ does not contain any cycle whose length be an odd number greater than $1$ and less than $ 2k + 1$. If $G$ has at most $n + \frac{(k-1) (n-1) (n+2)}{2}$ vertices, prove that the vertices of $G$ can be painted with $n$ colors in such a way that any edge of $G$ has its ends of different colors.