We give an Eynard-Orantin type topological recursion formula for the
canonical Euclidean volume of the combinatorial moduli space of pointed smooth
algebraic curves. The recursion comes from the edge removal operation on the
space of ribbon graphs. As an application we obtain a new proof of the
Kontsevich constants for the ratio of the Euclidean and the symplectic volumes
of the moduli space of curves.Comment: 37 pages with 20 figure
