## 1 Introduction

Bijective combinatorics of planar maps is a very active research topic, with several applications such as scaling limit results [24, 27, 1, 8], efficient random generation [32] and mesh encoding [28, 12]
(as well as giving combinatorial interpretations of beautiful counting formulas first discovered by Tutte [34, 35]
in the 60’s). In these bijections a map is typically encoded by a decorated tree structure.
Bijections for planar maps with no girth nor connectivity^{1}^{1}1The girth of a graph is the length of a shortest cycle in ; and is called -connected if one needs to delete at least vertices to disconnect it. conditions rely on geodesic labellings [31, 10]
and can be extended to any fixed genus [15, 13, 25] (where the map is encoded by a decorated unicellular map). On the other hand, when a girth condition or connectivity condition is imposed the bijections typically rely (see [2, 6] for general methodologies) on the existence of a certain ‘canonical’ orientation with prescribed outdegree conditions, such as Schnyder orientations
(introduced by Schnyder [33] for simple planar triangulations, and later
generalized to 3-connected planar maps [18] and to -angulations of girth [6, 7]), separating decompositions [16] or transversal structures [23, 19]; and it is not known how to specify such a canonical orientation in any fixed genus
(we also mention the powerful approach by Bouttier and Guitter [11] based on
decomposing so-called *slices*; this method, which yields a unified combinatorial decomposition for
irreducible maps with control on the girth and face-degrees, is yet to be extended to higher genus).

However, it has recently appeared that in the particular case of genus
(toroidal maps) a notion of canonical orientation amenable to a
bijective construction can be found, with the nice property that the
outdegree condition on vertices is completely homogenous (which is not
the case in the planar case, where the outdegree conditions differ for
vertices incident to the outer face). This has been done for
essentially simple^{2}^{2}2For a toroidal map , with
the corresponding infinite periodic planar representation, we say
that is essentially simple if is simple, and more
generally that has essential girth if has girth
, and we say that is essentially -connected if
is -connected. triangulations [17] (based
on earlier work on toroidal Schnyder woods [22], with graph drawing
applications) and more generally for maps with essential
girth and root-face degree [20], and for
essentially 4-connected triangulations [9]. It has also been
shown in [20] that, similarly as in the planar
case [6], such bijections (at least, the one in [20])
can be obtained by specializing a certain ‘meta’-bijection that is
derived from a construction due to Bernardi and Chapuy [5].

In this article we apply this methodology to the family of essentially 3-connected toroidal maps, where our bijection can be seen as the natural toroidal counterpart of the bijection developed in [21] between unrooted binary trees and so-called irreducible quadrangular dissections of the hexagon, which are closely related to 3-connected planar maps. Precisely, we describe in Section 3.1 a bijection (based on local closure operations very similar to those used in the planar case construction [21]) between a certain family of unicellular maps and a certain family of bipartite toroidal maps with one face of degree and all the other faces of degree (and having the property of being ‘essentially irreducible’, see Section 2 for definitions). This construction actually bears a strong resemblance with the recent bijection for essentially 4-connected toroidal triangulations described in [9] (this resemblance already appeared for the planar counterpart constructions given respectively in [21] and in [19]), and we conjecture that a unified bijection for ‘essentially irreducible’ toroidal -angulations should exist.

In Section 3.2 we then show that any rooted essentially -connected toroidal map can be decomposed (via the associated bipartite quadrangulation) into two parts by cutting along a certain ‘maximal’ hexagon: a planar part that can be treated by the planar case bijection [21], and a toroidal part that can be treated by our bijection . We then obtain in Section 3.3 a combinatorial derivation of the bivariate generating function of rooted essentially 3-connected toroidal maps counted by vertices and faces.

Similarly as in the planar case [21], the inverse bijection , which starts from and is described in Section 4, relies on canonical 3-biorientations (edges are either simply directed or bidirected, every vertex has outdegree ), to which the above-mentioned meta-bijection can be specialized. These biorientations are closely related to so-called balanced toroidal Schnyder orientations whose existence has been shown in [22, 26].

## 2 Preliminaries on maps

A *map* of genus is given by the embedding (up to
orientation-preserving homeomorphism) of a connected graph
(possibly with loops and multiple edges) on a compact orientable
surface , such that all components of are
homeomorphic to topogical disks, which are called the *faces* of
. The *genus* of is the genus of the underlying surface
; maps of genus and are called planar and toroidal,
respectively. Since the universal cover of the torus is the periodic
plane, a toroidal map can also be seen as an infinite periodic
planar map which we denote by , see Figure 1(a)
(in our figures, toroidal maps are drawn on the flat torus
, i.e., the unit square where the opposite
sides are identified; and the drawing lifts to a periodic planar
drawing upon replicating the square to tile the whole plane).

A *corner* of is the angular sector between two consecutive half-edges around a vertex.
The *degree* of a vertex or face of is the number of corners that are incident to it.
A map is called *rooted* if it has a marked corner, and is called *face-rooted* if it has a marked
face.
A *triangulation* (resp. *quadrangulation*) is a map with all faces of degree
(resp. of degree ). A map is called *6-quadrangular* if it has one face of degree
and all the other faces have degree (we will refer to the face of degree as the root-face).

A graph or map is called *bipartite* if its vertices are partitioned into black and white vertices,
such that every edge connects a black vertex to a white vertex.
For a map (whose vertices are white), the *angular map* of is the bipartite
map obtained by inserting a black vertex
inside each face of , and connecting to the vertices of at every corner around ,
and then deleting all the edges of (see Figure 1(b)).
It is well known that
the mapping from to is a bijection between maps of genus with vertices and faces,
and bipartite quadrangulations of genus with white vertices and black vertices.

On the other hand, the *dual* of is the map obtained
by inserting a vertex in each face of , and then for each edge of , with the
faces on each side of , drawing an edge accross that connects to ,
and finally deleting the vertices and edges of .
The *derived map* of is the map obtained by superimposing with (see Figure 1(c)). There are 3 types of vertices in : the vertices of are called *primal vertices*, the vertices of
are called *dual vertices*, and the vertices of degree at the intersection of each edge with its dual edge , which are called *edge-vertices*.
Note that is actually a quadrangulation. In fact, it is easy to see that it is
the angular map of the angular map of .
Note that , , and all have the same genus as .

A (possibly infinite) map is called *-connected* if it is simple and one needs to delete at least of its vertices to disconnect it.
A toroidal map is called *essentially -connected* if is -connected.
For instance the map in Figure 1(a)
is not -connected (since it has a double edge) but is essentially -connected^{3}^{3}3A visual characterisation is that a toroidal map is essentially -connected iff it admits a periodic planar straight-line drawing that is strictly convex (i.e., where all corners have angle smaller than ), which is satisfied in Figure 1(a)..
A (possibly infinite) map is called *irreducible* if its girth equals its minimum face-degree ,
and there is no cycle of length apart from face-contours.
A toroidal map is called *essentially irreducible* if is irreducible. Equivalently
in , every closed walk that delimits a contractible region on its right side has length at least (with the minimum face-degree),
and has length iff the enclosed region is a face.

In genus it is known that a map is 3-connected iff its angular map is irreducible. By an easy adaptation of the arguments to the periodic planar case (see also [30] for the higher genus case) one obtains:

###### Claim 1

For a toroidal map, is essentially -connected iff its angular map is essentially irreducible.

## 3 Main results

We let be the family of toroidal bipartite 6-quadrangular maps that are essentially irreducible and such that, apart from the root-face contour, there is no other closed walk of length enclosing a contractible region containing the root-face. We let be the family of toroidal bipartite quadrangulations that are essentially irreducible. We denote by the family of essentially -connected toroidal maps.

A map is called *unicellular*
if it has a unique face, and *precubic* if all its vertices have degree in .
In a precubic map, the vertices of degree are called *leaves* and those of degree
are called *nodes*; the edges incident to a leaf are called *pending edges*, the other ones are called
*plain edges*. We let be the family of precubic bipartite unicellular maps.
For we say that is *balanced* if any
(non-contractible) cycle of has the same number of incident edges on both sides, see
Figure 2(a) for an example.
Let be the set of balanced
elements in .

### 3.1 Bijection from to

We describe here a bijection from to , based on repeated closure operations, which can be seen as the toroidal counterpart of the bijection given in the planar case [21].

Let , and let be the number of leaves of . Then an application of the Euler formula easily implies
that has plain edges, hence has sides of plain edges. Initially the *root-face*
of is defined as its unique face. A clockwise walk around the root-face of (i.e., with the
root-face on our right) gives a cyclic word
on the alphabet ,
writing when we pass along a side of plain edge, and writing
when we pass along a leaf. Each time we see the pattern we can perform a so-called
*local closure* operation, which consists in merging the leaf at the end of the pending edge for
to the end of the edge-side corresponding to the third , forming a new quadrangular face,
see Figure 2(c). The resulting edge is now
considered as a plain edge in the new figure, and accordingly the pattern is replaced by
the letter in the cyclic word coding the situation around the root-face. We can then repeat
local closure operations (see Figure 2(d)),
each time decreasing by the numbers of ’s and by the number of ’s,
until there are no ’s. Note that the invariant is maintained all along (which also ensures
that there is at least one local closure possible at each step). Hence, at the end, the map we obtain
is a bipartite toroidal 6-quadrangular map. We let be the mapping that associates to (see
Figures 2(e) and 2(f)).

###### Theorem 1

The mapping is a bijection from to ; it preserves the number of black vertices and the number of white vertices.

### 3.2 Link between and essentially -connected toroidal maps

We let be the family of maps from with a marked edge, let be the family of maps from with a marked corner incident to a white vertex in the hexagonal face, and let be the family of rooted maps from (i.e., with a marked corner). Note that each corner in a map corresponds to an edge in the angular map. Hence, Claim 1 yields

(1) |

with the number of vertices (resp. faces) in the left-hand side corresponding to the number of white vertices (resp. black vertices) in the right-hand side.

We now explain how to decompose maps in into two parts, one planar and one in .

For , with its marked edge,
an *enclosing hexagon* of is a closed walk of length bounding a contractible region that
strictly contains . An enclosing hexagon is called *maximal* if its bounded
region is not included in the bounded region of another enclosing hexagon.

###### Lemma 2

Each , has a unique maximal enclosing hexagon.

*Proof.*
Note that the union of the two faces of incident to the marked edge
forms an enclosing hexagon, so the set of enclosing hexagons is non-empty, and thus there
is at least one maximal enclosing hexagon.

We first reformulate the definition of a hexagon. Let us define a
*region* of as given by where
are subsets of the vertex-set, edge-set and face-set of
, such that for all the edges incident to are in
, and for the faces incident to are in
. Note that the union (resp. intersection) of two regions is
also a region. We define a *boundary-edge-side* of as an incidence
face/edge of such that the face is in and the edge is not
in . The *boundary-length* of , denoted by , is
the number of boundary-edge-sides of . Since is bipartite,
the value of is even.
A *disk-region* is a
region homeomorphic to an open disk. An enclosing hexagon thus corresponds
to the (cyclic sequence of) boundary-edge-sides of a disk-region
such that and ; and it is *maximal* if there is no other
disk-region of boundary-length such that .
It is easy to see that for any two regions we have
(indeed any
incidence face/edge of has the same contribution to
as to ).

Suppose by contradiction that there are two distinct maximal enclosing hexagons, and let be the respective enclosed disk-regions. Since is essentially irreductible and both and contain , we have and . Moreover, , so . Thus is an enclosing hexagon, which contradicts the maximality of .

Let be the family of planar bipartite irreducible 6-quadrangular maps with at least one edge not incident to the hexagonal face, and let be the set of maps from with a marked edge not incident to the hexagonal face.

###### Claim 2

We have the following isomorphism:

(2) |

The number of black (resp. white) vertices in the left-hand side corresponds to three plus the total number of black (resp. white) vertices in the right-hand side

*Proof.*
For each we choose a vertex, denoted , among the white vertices incident to the hexagonal face (considered as the outer face of ).
To each pair we associate the bipartite quadrangulation obtained by patching within the
root-face of , with merged at the root-corner of . Clearly the contour of the root-face of becomes
the maximal enclosing hexagon of . We also have to check that is essentially irreducible, which amounts to check that there is no closed walk of length at most enclosing a contractible region that covers both faces in and in . An easy case-analysis ensures that it is not possible without creating a closed walk of length at most in that does not bound the hexagonal face but encloses a contractible region containing the hexagonal face, a contradiction.

Conversely, from , we let be the maximal enclosing hexagon of . Let be the map in formed by (unfolded into a simple 6-cycle) and the faces within the contractible region enclosed by . And let be the map obtained by emptying the area enclosed by ; the emptied area is now a hexagonal face where we mark the white corner where was.

The mappings from to and from to being clearly inverse of each other, we have a bijection between and .

### 3.3 Counting rooted essentially -connected toroidal maps

#### 3.3.1 Expression for the bivariate generating function

In this section we show that the bijection of Theorem 1, combined with the decomposition of Section 3.2 and the planar case bijection [21], allows us to derive an explicit algebraic expression for the (bivariate) generating function of rooted essentially -connected toroidal maps:

###### Theorem 3

Let be the generating function of , with and dual to the numbers of faces and vertices respectively. Then

(3) |

where and are the algebraic series specified by the system , and where and .

The initial terms of the series are

see Figure 3 below for an illustration (maps in with at most edges).

A rational expression in terms of actually exists in any genus.
Precisely, in genus we call a map *essentially -connected*
if it is -connected in the periodic representation
(in the Poincaré disk when ),
and let be the associated bivariate generating function with respect to the numbers of faces and vertices. In genus we call a quadrangulation *essentially irreducible* if
it has no closed walk of length enclosing a contractible region that is not a face.
The bijection of Claim 1 holds in any genus [30],
so that is also the series of edge-rooted bipartite essentially irreducible quadrangulations of genus ,
counted by black vertices and white vertices. It can be shown that has an explicit rational expression
in terms of and .
This can be done by a substitution approach (see e.g. [14])
that relates to the bivariate series of edge-rooted bipartite quadrangulations of genus , for which an explicit algebraic expression is known [4, 3]. Our
derivation (detailed in Section 3.3.3) is the first bijective one in genus
(a bijective derivation in genus is given in [21]).

#### 3.3.2 Univariate specializations

We mention here some univariate specializations of Theorem 3. First, note that is the generating function of by edges, the expression in Theorem 3 becomes

(4) |

The initial terms are . This is sequence A308524 in the OEIS.

On the other hand, is the generating function of by vertices, and the expression in Theorem 3 becomes

(5) |

The initial terms are . This is sequence A308526 in the OEIS.

We now consider the specialization to triangulations. Similarly as in genus , a toroidal triangulation is essentially 3-connected iff it is essentially simple. Note that in a toroidal essentially 3-connected map, all faces have degree at least , hence the numbers of vertices and faces satisfy , with equality iff the map is a triangulation. Hence if we let be the generating function of rooted essentially simple toroidal triangulations counted by vertices, then we have

We have , hence . If we let , then we have . Since dominates as ( being of order ), we find, as ,

Hence, from the expression in Theorem 3 we obtain

(6) |

The initial terms are . This is sequence A308523 in the OEIS. This expression of has also recently been derived bijectively in [20], and actually the bijection for triangulations given there can be seen as a specialization of the bijection we develop here, as we will see in Section 4.4.

#### 3.3.3 Proof of Theorem 3

The combinatorial proof of Theorem 3 that we present in this section can be seen as a bivariate adaptation of the proof of (6) given in [20] (itself an adaptation of the calculations in [15] for
bipartite quadrangulations of genus ).
A first remark is the combinatorial interpretation of the two series .
A *bipartite binary tree* is a planar bipartite precubic map. A black-rooted (resp. white-rooted) binary tree
is a bipartite binary tree with a marked pending edge incident to a black (resp. white) node.
Then clearly and are the generating functions of black-rooted and white-rooted binary trees,
counted with respect to the numbers of black nodes and white nodes. We also let and .

Let be the generating function of with (resp. ) dual to the number of black (resp. white) vertices that are not incident to the hexagonal face. Then it follows from [21] that is in bijection with bipartite binary trees with a marked edge (pending or plain), giving

We now let (resp. ) be the generating function of (resp. of ) with dual to the numbers of black and white vertices, respectively. Note that according to (1). Moreover, it follows from (2) that , hence we have

(7) |

It thus remains to compute . For a toroidal unicellular map ,
the *core* of is obtained from
by successively deleting leaves, until there is
no leaf (so has all its vertices of degree at
least ; the deleted edges form trees attached
at vertices of ). In we call *maximal chain*
a path whose extremities have degree larger than
and all non-extremal vertices of have degree .
Then the *kernel* of is obtained
from by replacing every maximal chain by an edge.
The kernel of a toroidal unicellular
map is either made of one vertex with two loops (double loop)
or is made of vertices and edges joining them
(triple edge). Note that in the first case, there is a vertex of
degree at least four. So this case never occurs for elements of
. Thus elements of have six half-edge in
their associated kernel. A map in is called *kernel-rooted* if
one of the six half-edges is marked.

Let be the generating function of kernel-rooted maps from with dual to the numbers of black and white vertices, respectively. The bijection of Theorem 1 and a classical double-counting argument ensure that , so that

(8) |

Hence it remains to express in terms of . Note that the generating function splits as

(9) |

depending on the colors of the two vertices of the kernel (with the one incident to the marked half-edge), and where the second equality follows from , since and play symmetric roles.

A *skeleton* is a toroidal unicellular precubic map such that every node belongs to the core.
Observe that a map is balanced if and only if its skeleton is balanced. We let
be the generating functions gathering the respective contributions from that are skeletons.
We clearly have

(10) |

A *bi-rooted caterpillar* is a bipartite binary tree with two marked leaves , called *primary root* and *secondary root*, such that every node is on the path from to . The *-score* of a bi-rooted caterpillar is the number of non-root leaves on the right-side of minus the number of non-root leaves
on the left-side of .

Clearly a skeleton (with a marked half-edge in the kernel) decomposes into an ordered triple of bi-rooted caterpillars, and is balanced if and only if the bi-rooted caterpillars have the same -score. Hence, if for we let , , be the generating functions of bi-rooted caterpillars of -score where are black/black (resp. black/white, white/white), and with (resp. ) dual to the number of black (resp. white) nodes, then we find

For , let be the number of walks of length with steps in , starting at and ending at (note that if ). We also define the generating function of walks ending at as

We clearly have for ,

We have for , and for we classically have (see [20] for more details)

where is the series of non-empty Dyck paths, and is the series of bridges ( is ), with dual to the half-length.

Hence we have

and replacing by we obtain

Similarly as in [15], we observe that is rational in , so it is also rational in since and Overall we obtain

Using (10) we deduce then

where we have used , , , .

## 4 The inverse bijection from to

### 4.1 Orientations

Let be a toroidal map endowed with an orientation .
For a non-contractible cycle of given with a
traversal direction, we denote by
(resp. ) the total number of edges going out of a
vertex on on the right (resp. left) side of , and we define the
*-score* of as .

For a graph and , an *-orientation* of (see [18]) is an orientation of such that every vertex has outdegree .
For a toroidal map, two -orientations of are
called *-equivalent* if every non-contractible cycle of
has the same -score in as in .

Assume is a face-rooted toroidal map, with its root face. An orientation
of is called *non-minimal* if there exists a non-empty set of faces
such that and every edge on the boundary of has a
face in on its right (and a face not in on its left). It is
called *minimal* otherwise.

###### Theorem 4 ([20])

Let be a face-rooted toroidal map that admits an -orientation . Then has a unique -orientation that is minimal and -equivalent to .

Moreover, for two -orientations of to be -equivalent, it is enough that and have the same -score on two non-contractible non-homotopic cycles of .

For an essentially -connected toroidal map, with its derived map,
a *Schnyder orientation* of is an orientation of such that every
primal or dual vertex has outdegree , and every edge-vertex has outdegree .
A Schnyder orientation is called *balanced* if for every non-conctractible cycle
not passing by dual vertices, one has .

The following result is shown in [26, Theorem 17, Lemma 8 and Proposition 11] (the algorithm to compute the orientation is based on repeated contraction operations done according to a careful case analysis):

###### Theorem 5 ([26])

Let be an essentially 3-connected toroidal map. Then admits a balanced Schnyder orientation.

Then a direct consequence of Theorem 4 is the following:

###### Theorem 6

Let be a face-rooted essentially 3-connected toroidal map. Then admits a unique minimal balanced Schnyder orientation.

An example is shown in Figure 5(b).

### 4.2 Biorientations

A *biorientation* of a map is map where each half-edge is oriented such that for each edge, at least one of its half-edges is outgoing. An edge is *bidirected* if both of its half-edges are outgoing.
The *outdegree* of a vertex is the number of outgoing half-edges incident to .
The *ccw-degree* of a face is the number of simply directed edges that have on their
left.
A *3-biorientation* is a biorientation where every vertex has outdegree . A 3-biorientation of a toroidal
quadrangulation, or of a toroidal 6-quadrangular map, is called *S-quad* if every quadrangular face
has ccw-degree . For a 6-quadrangular toroidal map, an easy argument based on the Euler
relation ensures that the ccw-degree of the root-face has to be in any S-quad 3-biorientation.

Let and let be the angular map of (we recall that is the angular map of ). For an S-quad 3-biorientation of , we let be the orientation of obtained by applying the local rules shown in Figure 4 (where is represented with red edges and with black edges).

The fact that all vertices have outdegree in implies that the primal and dual vertices have outdegree in . And the fact that every face of has ccw-degree implies that every edge-vertex in has outdegree . Finally the fact that the edges of have at least one outgoing half-edge implies that no face in is clockwise. Conversely, starting from a Schnyder orientation of with no clockwise face and applying the local rules one obtains an S-quad 3-biorientation of . To summarize:

###### Claim 3

The mapping is a bijection between the S-quad 3-biorientations of and the Schnyder orientations of with no clockwise face.

An example is shown in Figure 5(c).

An S-quad 3-biorientation of is called *balanced* if the corresponding Schnyder
orientation is balanced.

### 4.3 Description of the bijection

Let , and let be the bipartite quadrangulation obtained by adding a black vertex of degree inside the root-face of , and connecting to the corners at white vertices around , see Figure 5(a). If we let be the map in with a unique internal vertex that is black, then is obtained by patching within the root-face of , hence is in according to Claim 2.

Let be the map whose angular map is . Since is the angular map of , each face of
corresponds to an edge of . We choose (arbitrarily) the root-face among the faces of
incident to . We let be the minimal balanced
Schnyder orientation of , see Figure 5(b).
Since is a source in , the root-face contour is not a clockwise cycle. In addition
the contours of the other faces are neither clockwise cycles by minimality. Hence has no clockwise face.
The fact that is a source also implies that is minimal for any of the root-face choices, hence does not depend on which of the faces incident to is chosen.
Let be the associated S-quad 3-biorientation of (obtained using the rules of Figure 4).
We let be the biorientation of , called the *canonical biorientation* of , which is obtained from by deleting and its 3 incident edges. We will see in Section 5.3.2 (Lemma 15) that in , the 3 edges at are simply directed (out of ), hence is an S-quad 3-biorientation.

We can now describe the mapping . For endowed with its canonical biorientation , we simply let be the map obtained by deleting all the ingoing half-edges (thus any simply directed edge is turned into a pending edge), see Figure 5(c)-(d).

###### Theorem 7

The mapping is a bijection from to ; it preserves the number of black vertices and the number of white vertices.

We prove this result in Section 5.

### 4.4 Specialization to triangulations

As already mentioned, a toroidal triangulation is essentially 3-connected iff it is
*essentially simple*, i.e., is simple. Let
be the family of face-rooted essentially simple toroidal triangulations such that, apart from
the root-face contour, there is no other closed walk of length enclosing a contractible
region that contains the root-face.
Let (resp. ) be the subfamily of (resp. )
where the respective numbers of white vertices
and black vertices satisfy . For this amounts to having all black vertices of degree ,
and for this amounts to having no pending edge incident to a white node.
The bijection of Theorem 7 specializes into a bijection between and .
For , we let be its angular map, with
the black vertex corresponding to the root-face of , and let be the map obtained from
by deleting and its 3 incident edges. Then it can be checked that and that gives
a bijection between and . Thus the specialization of yields a bijection
between and ; we actually recover here the bijection between and
recently given in [20].

## 5 Proof of Theorem 7

### 5.1 Bijection between right biorientations and bimobiles

Let be a map endowed with a biorientation such that every vertex has at least one outgoing
half-edge. For an outgoing half-edge of , we define the
*rightmost walk* from as the (necessarily unique and
eventually looping) sequence of half-edges starting from ,
at each step taking the opposite half-edge and then the rightmost
outgoing half-edge at the current vertex; in other words,
for each , is opposite to , and is the next outgoing
half-edge after in counterclockwise order around their incident vertex.

If is face-rooted, a biorientation of is called a *right biorientation*
if every vertex has at least one outgoing half-edge, and for every outgoing half-edge , the rightmost walk starting
from eventually loops on the contour of the root-face with on
its right side.
For ,
denotes the family of right biorientations
of toroidal face-rooted maps whose root-face has degree .

We call *(toroidal) bimobile* a toroidal unicellular
map with two kinds of vertices, round or square, with no square-square edge, and such that each
corner at a square vertex might carry additional dangling half-edges
called *buds*. The *excess* of a bimobile is the number of
round-square edges plus twice the number of round-round edges, minus the
number of buds. We let be the family of bimobiles of excess .

For (whose vertices are considered round) we denote by the embedded graph obtained by inserting a square vertex in each face of , then applying the local rules of Figure 6 to every edge of (thereby creating some edges and buds), and finally erasing the isolated square vertex in the root-face of (since the orientation is right, this square vertex is incident to buds and no edge), see Figure 7 for an example.

The following result has been obtained in [20], derived from the bijection for covered maps given in [5]:

###### Theorem 8 ([20, Theorem 14])

For , the mapping is a bijection between the family and the family . Each vertex of outdegree in becomes a round vertex of degree in , and each non-root face of degree and ccw-degree in becomes a black vertex of degree with neighbours (and buds) in .

### 5.2 Specialization to S-quad 3-biorientations

We let be the family of S-quad 3-biorientations where the underlying toroidal map is bipartite 6-quadrangular, and the biorientation is in . On the other hand, we let be the family of bimobiles of excess such that the round vertices are partitioned into black and white vertices without two white round vertices or two black round vertices adjacent, where all round vertices have degree and all square vertices have degree with incident buds. Note that and are easily in bijection: indeed each can be turned into a bimobile , upon replacing every leaf by a square vertex with buds attached (Euler’s formula and the fact that is precubic easily imply that must have excess ).

###### Proposition 9

The mapping induces a bijection between and , hence a bijection between and . The bijection from to amounts to the deletion of the ingoing half-edges.

*Proof.*
Let be defined as except that no vertex-bipartition is required. Similarly, let
be defined as except that no bipartition of the round vertices is required. Clearly
induces a bijection between and . Moreover, if the vertices in are bipartitioned
(the underlying map is bipartite) then it induces a bipartition of the round vertices in so that
every round-round edge has a white extremity and a black extremity (indeed such an
edge already exists in the bidirected map). It remains to show that if the round
vertices of can be bipartitioned, then it induces a bipartition of the vertices of the associated
bidirected map . The crucial point is that since is a toroidal map (with one face) and since the square vertices
are leaves, the embedded graph obtained from by deleting all pending edges is a toroidal map. Hence
there are two non-homotopic non-contractible cycles in . Since the colors alternate along each cycle, these two cycles have even length. Note also that these two cycles are also present in the bidirected map (they are cycles of bidirected edges), hence has two non-homotopic non-contractible cycles of even length. Since the faces of
have even degree, we conclude that has a valid vertex-bipartition (which is unique up to the choice of the color of a given vertex). Note also that is a connected spanning submap of . Hence the vertex-bipartition of is a valid vertex-bipartition for (such that no two adjacent vertices have the same color), so that inherits bipartiteness from .
Finally the fact that the bijection from to amounts to deleting the ingoing half-edges is
shown in Figure 8.

###### Lemma 10

Let be a toroidal bipartite 6-quadrangular map. If can be endowed with a biorientation , then .

*Proof.*
Let be a contractible closed walk of length in , such that there is at least one vertex
strictly inside (i.e., not on ) the enclosed region . Let and be respectively the numbers of vertices and edges strictly inside . The Euler relation ensures that where if does not contain the root-face , and if contains . We let be the mobile associated to ,
and let be the part of restricted to the round vertices strictly inside and to the square vertices
for the faces in ; in we do not retain the three buds at each square vertex.
Note that is a forest (if had a cycle,
it would yield a contractible cycle in , a contradiction) where the round vertices have degree and the square vertices have degree in . It has nodes,
and from the local conditions of one easily sees that the number of edges in is equal
to the number of bidirected edges with both ends strictly inside plus the number of
simply directed edges starting from a vertex strictly inside . In particular
has at most edges. On the other hand, the degree conditions imply that its number of edges is ,
with the number of connected components (trees) not reduced to a single square vertex.
We conclude that , hence if does not contain
and if contains . This easily ensures that has no contractible 2-cycle (since
there is no face of degree , such a 2-cycle would have to strictly enclose at least one vertex) nor a non-facial contractible 4-cycle, nor a contractible 6-cycle enclosing and different from the contour of .

###### Lemma 11

Every map has a unique biorientation in such that the associated (by Proposition 9) unicellular map is in . This biorientation is actually the canonical biorientation of .

### 5.3 Proof of Lemma 11

#### 5.3.1 Properties of 3-biorientations

In this section we let , and let be the map whose angular map is . We let be an S-quad 3-biorientation of . For a closed walk on that encloses a region homeomorphic to an open disk on its right, we let (resp. ) be the number of outgoing half-edges of that are encountered just after (resp. just before) a vertex while walking along around , and let be the number of outgoing half-edges of that are in the interior of and incident to a vertex of .

###### Lemma 12

If has length , then .

*Proof.*
Consider the map obtained from by keeping all the
vertices and edges that lie in the region , including . The
vertices that appear several times on are repeated,
so is consider as a planar map whose outer face boundary is turned into a simple cycle.
In we call *inner* the vertices and edges not incident to the outer face, and call *inner faces*
those different from the outer face.
Let