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We give an Eynard-Orantin type topological recursion formula for thecanonical Euclidean volume of the combinatorial moduli space of pointed smoothalgebraic curves. The recursion comes from the edge removal operation on thespace of ribbon graphs. As an application we obtain a new proof of theKontsevich constants for the ratio of the Euclidean and the symplectic volumesof the moduli space of curves.

The Kontsevich constants for the volume of the moduli of curves and topological recursion