Unified bijections for maps with prescribed degrees and girth

Olivier Bernardi (1) , Eric Fusy (2)

1 MIT - Massachusetts Institute of Technology
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

This article presents unified bijective constructions for planar maps, with control on the face degrees and on the girth. Recall that the girth is the length of the smallest cycle so that maps of girth at least $d=1,2,3$ are respectively the general, loopless, and simple maps. We first present bijections for planar maps with one root-face (plane maps) of degree $d$ equal to the girth. We then present more general bijective constructions for maps with two root-faces (annular maps) with girth at least $d$ and root-faces of arbitrarily fixed degrees. Our bijections associate a decorated plane tree to each map, and non-root faces of degree $k$ of the map correspond to vertices of degree $k$ of the tree. As special cases we recover several known bijections for bipartite maps, loopless triangulations, simple triangulations, simple quadrangulations, etc. In terms of counting, we obtain for each integer~$d$ an expression of the generating function $F_d(x_d,x_{d+1},x_{d+2},\ldots)$ of rooted plane maps of girth~$d$ with root-face of degree $d$, where the variable~$x_k$ marks the number of non-root faces of degree~$k$. The expression for~$F_1$ was already obtained (bijectively) by Bouttier, Di~Francesco and Guitter, but for $d\geq 2$ the expression of~$F_d$ is new. We also obtain an expression for the generating function $\G_{p,q}^{(d,e)}(x_d,x_{d+1},\ldots)$ of rooted annular maps with root-faces of degrees $p$ and $q$ such that cycles separating the two root-faces have length at least $e$ while other cycles have length at least $d$, where the variable $x_k$ marks the number of non-root faces of degree $k$. Our strategy is to obtain all the bijections as specializations of a single ''master bijection'' introduced by the authors in a previous article. The master bijection sends a class $\mathcal{O}$ of oriented maps to a class of decorated trees, and the core of our approach is to exhibit certain ''canonical orientations'' for maps with prescribed girth constraints so as to identify these classes of maps with subclasses of $\mathcal{O}$ on which the master bijection specializes nicely.

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Eric Fusy. Unified bijections for maps with prescribed degrees and girth. 2011. ⟨hal-00566762v2⟩