  In this article, we discuss some classical problems in combinatorics which
can be solved by exploiting analogues between graph theory and the theory of
manifolds. One well-known example is the McMullen conjecture, which was settled
twenty years ago by Richard Stanley by interpreting certain combinatorial
invariants of convex polytopes as the Betti numbers of a complex projective
variety. Another example is the classical parallel redrawing problem, which
turns out to be closely related to the problem of computing the second Betti
number of a complex compact $(\C^*)^n$-manifold.
