Abstract. \begin{abstract} \nin We present a direct proof of Kolmogorov's theorem on the persistence of quasi-periodic solutions for nearly integrable, real--analytic Hamiltonian systems with Hamiltonians of the form $\frac{1}{2} y\cdot y-\e f(x)$ where $(y,x)$ $\in$ $\rN \times\tN$ are standard symplectic coordinates. The method of proof consists in constructing, via graph theory, the formal solution as a formal power series in $\e$ and to show that the $k^{\rm th}$ coefficient of such formal series can be bounded by a constant to the $k^{th}$ power. All details are presented in a self contained way (included what is needed from the theory of graphs). \end{abstract}