Paper Info
Title
Orientations and bijections for toroidal maps with prescribed face-degrees and essential girth
Abstract
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, 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 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.
Year
DOI
Venue
2018
10.1016/j.jcta.2020.105270
Journal of Combinatorial Theory, Series A
Keywords
Field
DocType
Toroidal maps,d-angulations,Girth constraints,Orientations,Bijections,Generating functions
Discrete mathematics,Generating function,Combinatorics,Bijection,Bipartite graph,Torus,Bijection, injection and surjection,Toroid,Algebraic expression,Periodic graph (geometry),Mathematics
Journal
Volume
ISSN
Citations 
175
0097-3165
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Éric Fusy119821.95
Benjamin Lévêque29814.12