Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth
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We present unified bijections for maps on the torus with control on the face-degrees and essential girth (girth of the periodic planar representation). A first step is to show that for d>=3 every toroidal d-angulation of essential girth d can be endowed with a certain "canonical" orientation (formulated as a weight-assignment on the half-edges). Using an adaptation of a construction by Bernardi and Chapuy, we can then derive a bijection between face-rooted toroidal d-angulations of essential girth d (with the condition that no d-angle encloses the root-face apart from its contour) and a family of decorated unicellular maps. The orientations and bijections can then be generalized to toroidal face-rooted maps of essential girth d (for any d>= 1) with a root-face of degree d (and again no d-angle enclosing the root-face apart from its contour), and they take a simpler form in the bipartite case, as a parity specialization. On the enumerative side we obtain explicit algebraic expressions for the generating functions of rooted essentially simple triangulations and bipartite quadrangulations on the torus. Our bijective constructions can be considered as toroidal counterparts of those obtained by Bernardi and the first author in the planar case, and they also build on ideas introduced by Despré, Gonçalves and the second author for essentially simple triangulations, of imposing a balancedness condition on the orientations in genus 1.
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