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The BKMP conjecture (2006-2008), proposed a new method to compute closed andopen Gromov-Witten invariants for every toric Calabi-Yau 3-folds, through atopological recursion based on mirror symmetry. So far, this conjecture hadbeen verified to low genus for several toric CY3folds, and proved to all genusonly for C^3. In this article we prove the general case. Our proof is based onthe fact that both sides of the conjecture can be naturally written in terms ofcombinatorial sums of weighted graphs: on the A-model side this is thelocalization formula, and on the B-model side the graphs encode the recursivealgorithm of the topological recursion. One can slightly reorganize the set ofgraphs obtained in the B-side, so that it coincides with the one obtained bylocalization in the A-model.Then it suffices to compare the weights of verticesand edges of graphs on each side, which is done in 2 steps: the weightscoincide in the large radius limit, due to the fact that the toric graph is thetropical limit of the mirror curve. Then the derivatives with respect toK\"ahler radius coincide due to special geometry property implied by thetopological recursion.

Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture