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This paper is concerned with the design of a high-order discontinuous Galerkin (DG) method for solving the 2-D time-domain Maxwell equations on nonconforming triangular meshes. The proposed DG method allows for using nonconforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme. Numerical experiments are presented which both validate the theoretical results and provide further insights regarding to the practical performance of the proposed DG method, particulary when nonconforming meshes are employed.