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