A bijection for essentially 3-connected toroidal maps

We present a bijection for toroidal maps that are essentially 3-connected (3-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite with all faces of degree 4 except hexagonal root-face. show these well-characterized unicellular maps. bijection, to recent one by Bonichon and Lévêque 4-connected triangulations, can be seen as counterpart developed case Fusy, Poulalhon Schaeffer, it extends recently proposed Fusy simple triangulations. Moreover, we rooted decomposed into two pieces, part is treated our above-mentioned bijection. This yields combinatorial derivation bivariate generating function maps, counted vertices faces.