A short proof of the lambda_g-conjecture without Gromov-Witten theory:   Hurwitz theory and the moduli of curves
  We give a short and direct proof of the $\lambda_g$-Conjecture. The approach is through the Ekedahl-Lando-Shapiro-Vainshtein theorem, which establishes the ``polynomiality'' of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of Gromov-Witten theory.   We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures. 
Ian P. Goulden, David M. Jackson, and Ravi Vakil