  Asymptotic expansions of Gaussian integrals may often be interpreted as
generating functions for certain combinatorial objects (graphs with additional
data). In this article we discuss a general approach to all such cases using
colored graphs. We prove that the generating power series for such graphs
satisfy the same system of partial differential equations as the Gaussian
integral and the formal power series solution of this system is unique. The
solution is obtained as the genus expansion of the generating power series. The
initial term of this expansion is the corresponding generating function for
trees. The consequence equations for this term turns to be equivalent to the
inversion problem for the gradient mapping defined by the initial condition.
The equations for the higher terms of the genus expansion are linear. The
solutions of these equations can be expressed explicitly by substitution of the
initial conditions and the initial term (the tree expansion) into some
universal polynomials (for g>1) which are generating functions for stable
closed graphs. (For g=1 instead of polynomials appears logarithm.) The stable
graph polynomials satisfy certain recurrence. In [1] some of these results were
obtained for the one dimensional case by more or less direct solution of
differential equations. Here we present purely combinatorial proofs.
