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.MSC Primary: 14N35, 05C30, 53D30, 11P21; Secondary: 81T30 1. The Witten-Kontsevich theory for the tautological cotangent class (i.e. the ψ-class) 2 The combinatorial model of the moduli space Let us begin with reviewing basic facts about ribbon graphs and the combinatorial model of the moduli space M g,n due to Harer [22], Mumford [38], and Strebel [47]. We refer to [35] for precise definitions and more detailed exposition.A ribbon graph of topological type (g, n) is the 1-skeleton of a cell-decomposition of a closed oriented topological surface Σ of genus g that decomposes the surface into a disjoint union of v 0-cells, e 1-cells, and n 2-cells. The Euler characteristic of the surface is given by 2 − 2g = v − e + n. The 1-skeleton of a cell-decomposition is a graph Γ drawn on Σ,

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