Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth

International audienceWe 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, apart from the root-face contour, no other closed walk of length d encloses the root-face) and a family of decorated unicellular maps. The orientations and bijections can then be generalized, for any d ≥ 1, to toroidal face-rooted maps of essential girth d with a root-face of degree d (and with the same root-face contour condition as for d-angulations), 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 trian-gulations 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

On polynomials counting essentially irreducible maps

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess contractible cycles of length less than $2b$ and each such cycle of length $2b$ is required to bound a face of degree $2b$. The enumeration of such maps is shown to be again given by a symmetric polynomial $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ in the face degrees with a polynomial dependence on $b$. These polynomials satisfy (generalized) string and dilaton equations, which for $g\leq 1$ uniquely determine them. The proofs rely heavily on a substitution approach by Bouttier and Guitter and the enumeration of planar maps on genus-$g$ surfaces.Comment: 37 pages, 5 figure