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

07/02/2018 ∙ by Éric Fusy, et al. ∙ 0

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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1 Introduction

The enumerative study of (rooted) maps has been a very active research topic since Tutte’s seminal results on the enumeration of planar maps [37, 38], later extended to higher genus by Bender and Canfield [4]. Tutte’s approach is based on so-called loop-equations for the associated generating functions with a catalytic variable for the root-face degree. Powerful methods have been developed to compute the solution of such equations (originally solved by guessing/checking), both in the planar case [28, 13] and in higher genus [22].

The striking simplicity of counting formulas discovered by Tutte (e.g., the number of rooted planar simple triangulations with vertices is equal to ) asked for bijective explanations. The first such constructions, bijections from maps to certain decorated trees, were introduced by Cori and Vauquelin [20] and Arquès [3] and later further developed by Schaeffer [35], who also introduced with Marcus the first bijection (for bipartite quadrangulations) that extends to higher genus [35, Chap.6]. The bijection has been adapted in [19] to a form better suited for computing the generating functions, and has been recently extended to non-orientable surfaces [17, 11].

In the planar case many natural families of maps considered in the literature are given by restrictions on the face-degrees and on the girth (length of a shortest cycle). For instance loopless triangulations are (planar) maps with all face-degrees equal to and girth at least . The bijections developed over the years for such families (in particular, simple quadrangulations [35, Sect.2.3.3], loopless triangulations [35, Sect.2.3.4], simple triangulations [32], irreducible quadrangulations [25] and triangulations [24]) shared the feature that each map of the considered family can be endowed with a ‘canonical’ orientation that is usually specified by outdegree prescriptions (so-called -orientations [23]), which is then exploited to associate to the map a decorated tree structure. For instance simple triangulations with a distinguished outer face can be endowed with an orientation where all outer vertices have outdegree and all inner vertices have outdegree , such orientations being closely related to Schnyder woods [36]. In recent works [7, 2] the methodology has been given a unified formalism, where each such bijective construction can be obtained as a specialization of a ‘meta’-bijection between certain oriented maps and certain decorated trees, which itself is an adaptation of a bijection developed in [5] (and extended in [6] to higher genus) to count tree-rooted planar maps. A success of this strategy has been to solve for the first time [8] the problem of counting planar maps with control on the face-degrees and on the girth (this has been subsequently recovered in [15] and extended to the so-called irreducible setting), and to adapt the bijections to hypermaps [9] and maps with boundaries [10].

Up to now this general strategy based on canonical orientations has been mostly applied in the planar case, while the only bijections known to extend to any genus deal with maps (or bipartite maps) with control on the face-degrees but not on the girth: bijections to labeled mobiles [14, 19, 16] or to blossoming trees and unicellular maps [34, 29]. It has however recently appeared [21] that in the case of genus , a bijection based on canonical orientations can be designed for essentially simple triangulations111A map on the torus is said to have ‘essentially’ property if the periodic planar representation of has property ; thus is essentially simple means that is simple. Similarly the essential girth of is defined as the girth of . The canonical orientations used in this construction are -orientations (all vertices have outdegree ) with an additional ‘balancedness’ property (every non-contractible cycle has the same number of outgoing edges to the left side as to the right side), see Figure 1(a) for examples. The existence of such orientations builds on an earlier work on toroidal Schnyder woods [27] (see also [30]), and the obtained bijection can be considered as a toroidal counterpart of the one in [32]. This strategy has also been recently applied to essentially 4-connected triangulations [12], where the obtained bijection (based on certain ‘balanced’ transversal structures) is now a toroidal counterpart of the one in [24].

Figure 1: (a) Examples of -orientations for a toroidal essentially simple triangulation (the first example is not balanced as the bold cycle has outdegree on the upper side and outdegree on the lower side, the second example is balanced). (b) Examples of -orientations for a toroidal pentagulation of essential girth (the first example is not balanced as the bold cycle has total weight on the upper side and total weight on the lower side, the second example is balanced)

Main results and outline of the article.

In this article, we extend the strategy of [21] to toroidal maps of prescribed essential girth and face-degrees, thereby obtaining bijections to certain decorated unicellular maps. Our bijections can be seen as toroidal counterparts of those given in [7] for planar toroidal -angulations of essential girth ) and in [8] for planar maps with prescribed girth and face-degrees.

Our first results deal with toroidal -angulations of essential girth , for . In the planar case it is known [7] that -angulations of girth , with a marked face considered as the outer face, can be endowed with certain ‘weighted biorientations’ (given by assigning a weight in to every half-edge) called -orientations, such that for every inner edge (resp. inner vertex) the sum of the weights of the incident half-edges is (resp. ). Moreover, each -angulation of girth admits a ‘canonical’ such orientation, called the minimal one. The meta-bijection given in [7] can then be applied to the minimal -orientations, giving a correspondence with well-characterized decorated trees.

We will prove that a parallel strategy can be applied in genus . Precisely, we show in Section 3 that every toroidal -angulation of essential girth admits a so-called balanced -orientation, where again every half-edge is assigned a weight-value in such that the total weight of each edge (resp. vertex) is (resp. ) and ‘balanced’ means that for every non-contractible cycle , the total weight of half-edges incident to each side of is the same, see Figure 1 for examples ( on the left side, on the right side). Similarly as in the planar case, when the -angulation has a distinguished face, the map admits a ‘canonical’ such orientation, called the minimal one. An extension of the ‘meta-bijection’ to higher genus (described in Section 4.2 and obtained by adapting the construction of [6]) can then be applied to these orientations, yielding a bijection, stated in Section 4.5, between face-rooted toroidal -angulations of essential girth , with the extra condition that there is no -angle enclosing the root-face apart from its contour, and a family of well-characterized decorated unicellular maps of genus .

Similarly as in the planar case [8], the strategy can then be extended to face-rooted toroidal maps of essential girth , with root-face degree (again with the condition that there is no -angle enclosing the root-face apart from its contour). The canonical orientations in that case have similar weight conditions, now allowing for half-edges of negative weights, and the obtained bijections, stated in Section 4.6, keep track of the distribution of the face-degrees, and have a simpler form in the bipartite case (which can be seen as a parity specialization of the general bijection, as in the planar case [7, 8]).

While the corresponding unicellular decorated trees are well-characterized, we have not succeeded to develop a strategy to get a manageable expression of the associated generating function for arbitrary values of . However we show in Section 5 that in certain cases (essentially simple triangulations and essentially simple bipartite quadrangulations), the computations can be carried by a similar approach as in [19], and the expressions simplify nicely.

Higher genus extensions?

It is unclear to us if our results could be extended to higher genus. The nice property of the torus is that the Euler characteristic is zero, which is compatible with orientations having homogeneous outdegrees (e.g. for triangulations on the torus there are exactly times more edges than vertices, and the orientations exploited to derive a bijection are those with outdegree at each vertex).

In higher genus it has been shown in [1] that every simple triangulation has an orientation where every vertex-outdegree is a nonzero multiple of , hence all vertices have outdegree except for special vertices whose outdegree is a multiple of larger than (e.g. in genus all vertices have outdegree except for either two vertices of outdegree or one vertex of outdegree ), and the presence of these special vertices makes it more difficult to come up with a natural canonical orientation amenable to a bijection.

2 Preliminaries

A map of genus is an embedding of a connected graph (possibly with loops and multiple edges) on the orientable surface of genus , such that all components of are homeomorphic to open disks; we will mostly consider maps of genus , which we call toroidal maps. A map is called rooted if it has a marked corner, and is called face-rooted if it has a marked face. The dual of is the map obtained by inserting a vertex in each face of , every edge yielding a dual edge in that connects the vertices dual to the faces on each side of . A walk in is a (possibly infinite) sequence of edges traversed in a given direction, such that the head of an edge in the sequence coincides with the tail of the next edge in the sequence (possibly two successive edges in the sequence are the same edge traversed in opposite directions). A path in is a walk with no repeated vertices. A closed walk in is a finite walk such that the head of the first edge in the sequence coincides with the tail of the last edge. A cycle is a closed walk with no repeated vertices.

The girth of a map is the length of a shortest cycle in . The essential girth of a toroidal map is the girth of the universal cover (periodic planar representation). Hence the essential girth is at least the girth. A contractible closed walk of is defined as a closed walk having a contractible region on its right, which is called the interior of . The essential girth can equivalently be defined as the length of a shortest contractible closed walk (indeed such a walk lifts to a cycle in , and a shortest cycle in is a lift of a contractible closed walk). If has essential girth , a -angle of is a contractible closed walk of length ; it is called maximal if its interior is not contained in the interior of another -angle. A toroidal map is called essentially simple if it has essential girth at least (it means that is simple, i.e., has no loop nor multiple edges).

For , a map is called a -angulation if all its faces have degree . For , such maps are respectively called triangulations, quadrangulations, pentagulations. Note that a toroidal -angulation has essential girth less or equal to (and it can be strictly less). A toroidal -angulation of essential girth is called a -toroidal map. Note that -toroidal maps are exactly essentially simple toroidal triangulations. By Euler’s formula, one can check that, in a toroidal map with all face-degrees even, a contractible closed walk must have even length. In particular, -toroidal maps are exactly essentially simple quadrangulations.

For a map with vertex-set and edge-set , and , an -orientation [23] of is an orientation of such that every vertex has outdegree . A biorientation of is the assignment of a direction to every half-edge (half-edges can be either outgoing or ingoing at their incident vertex). The outdegree of a vertex is the total number of outgoing half-edges incident to . An -biorientation of is a biorientation of where every half-edge is given a value in , which is if the half-edge is outgoing and equal to zero if the half-edge is ingoing. The weight of a vertex is the total weight of its incident half-edges. The weight of an edge is the total weight of its two half-edges. Note that an orientation can be identified to an -biorientation where every edge has weight . For and an -orientation of is an -biorientation of such that every vertex has weight and every edge has weight . In all this paper, we assume that takes only values. By doing so we can define the -expansion of as the map obtained from after replacing every edge of by a group of parallel edges connecting and . Note that every -orientation of yields an -orientation of , see Figure 2.(a). Conversely every -orientation of yields an -orientation of , called the -expansion of , with the convention that the edge-directions in the group of parallel edges are chosen in the unique way consistent with the weights and such that there is no clockwise cycle within the group, as shown in Figure 2.(b).

Figure 2: (a) Rule to obtain an -orientation of from an -orientation of (with the -expansion of ). (b) Rule to obtain an -orientation of from an -orientation of .

Assume is a face-rooted map, with its marked 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. An -orientation of is called minimal if its -expansion is minimal (where the root-face of is the one corresponding to ).

Consider an orientation of and a non-contractible cycle of given with a traversal direction. Let (resp. ) be the number of edges of crossing from left to right (resp. from right to left). Then the -score of is defined as . Two -orientations are called -equivalent if every non-contractible cycle of has the same -score in and in . The following statement is easily deduced from the results and observations in [33]:

Theorem 1 ([33])

Let be a face-rooted map on the orientable surface of genus endowed with an -orientation . Then has a unique -orientation that is minimal222It is actually proved in [33] that the set of -orientations that are -equivalent to is a distributive lattice, of which is the minimum element. and -equivalent to .

Moreover, suppose that is a toroidal map and are two -orientations of . If two non-contractible non-homotopic333Two closed curves on a surface are called homotopic if one can be continuously deformed into the other. cycles of have the same -score in and in , then are -equivalent.

We now define the analogue of the function introduced in [21, 26] for Schnyder woods (see also [30] for a detailed presentation).

If is endowed with an orientation, and is 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 define the -score of as . Two -orientations of are called -equivalent if every non-contractible cycle of has the same -score in as in . The following theorem is an analog (and a consequence) of Theorem 1; we only state it in genus , to keep the proof simpler and as it is the focus of the article.

Theorem 2

Let be a face-rooted toroidal map endowed with 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 two non-contractible non-homotopic cycles of have the same -score in and in .

Proof.  The completion-map of is the map obtained by superimposing and . The vertices of are of 3 types: primal vertices (those of ), dual vertices (those of ) and edge-vertices (those, of degree , at the intersection of an edge with its dual edge ). Let be the function from the vertex-set of to such that, if is a primal vertex of then , if is a dual vertex of then , and if is an edge-vertex of then . Note that any -orientation of yields an -orientation of : each edge of corresponding to a half-edge of an edge is assigned the direction of in , and each edge of corresponding to a half-edge of an edge is directed toward the incident edge-vertex. Clearly the mapping sending to is a bijection from the -orientations of to the -orientations of , with the property that is minimal if and only if is minimal.

Let be a non-contractible cycle of given with a traversal direction. Let be the cyclic sequence of corners of that are encountered when walking “just to the right” of . Since every corner of corresponds to a face of , the cyclic sequence identifies to a non-contractible cycle of , which we denote by , see Figure 3 (note that is clearly homotopic to ). It is then easy to see that for every -orientation of , we have

Hence, for two -orientations of and a non-contractible cycle of given with a traversal direction, we have iff iff iff . Hence are -equivalent if and only if are -equivalent, where we use the second statement in Theorem 1 to have the ‘only if’ direction444While we do not need it here, we also mention that it is easy to prove by similar arguments that are -equivalent iff are -equivalent. Hence the -equivalence classes on -orientations are the same as the -equivalence classes on -orientations (which are distributive lattices)..

Figure 3: Left: a non-contractible cycle of . Middle: the situation in in the right neighborhood of . Right: the corresponding non-contractible cycle of (which passes by vertices, represented as triangles, that are in the faces corresponding to the corners incident to on its right side).

It is then easy to prove the theorem. For an -orientation of , Theorem 1 ensures that there exists an -orientation of that is minimal and -equivalent to . By what precedes, is -equivalent to (and is minimal), hence we have the existence part. Moreover, if there was another -orientation minimal and -equivalent to , then would be minimal, -equivalent to , and different from , yielding a contradiction. This gives the uniqueness part.

We now prove the second statement of the theorem. Let be two -orientations of that have the same -score for two non-contractible non-homotopic cycles . By what precedes, and have the same -score in and in . Hence, by Theorem 1, and are -equivalent, so that and are -equivalent.

More generally if is endowed with an -biorientation and is a non-contractible cycle of given with a traversal direction, we denote by (resp. ) the total weight of half-edges incident to a vertex on on the right (resp. left) side of , and define the -score of as . We say that an -biorientation of is balanced if the -score of any non-contractible cycle of is .

Two -orientations, are called -equivalent if every non-contractible cycle of has the same -score in and in . The following theorem is a generalization (and a consequence) of Theorem 2 that will be useful for our purpose.

Theorem 3

Let be a face-rooted toroidal map endowed with 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 two non-contractible non-homotopic cycles of have the same -score in and in .

Proof.  Let be the -expansion of . For an -orientation of , let be the -expansion of , i.e., the -orientation of obtained from by applying the rule of Figure 2(b). For a non-contractible cycle of given with a traversal direction, let be the non-contractible cycle of that goes along in the “rightmost” way, i.e. for each edge of , the cycle passes by the rightmost edge in the group of edges arising from . Clearly

Hence, for two -orientations of and for a non-contractible cycle of given with a traversal direction, we have iff iff iff . Hence are -equivalent if and only if are -equivalent (we use the second statement in Theorem 2 to have the ‘only if’ direction).

For an -orientation of , Theorem 2 ensures that has an -orientation that is minimal and -equivalent to . Let be the -orientation obtained from by applying the rule of Figure 2(a). Since is minimal, there is no clockwise cycle inside any group of edges associated to an edge . Hence is the -expansion of , so that (by definition) is minimal, and moreover it is -equivalent to . This proves the existence part. If there was another -orientation minimal and -equivalent to , then would be minimal, -equivalent to , and different from , contradicting Theorem 2. This proves the uniqueness part.

Let us now prove the second statement of the theorem. Let be two -orientations of that have the same -score for two non-contractible non-homotopic cycles . By what precedes, and have the same -score in and in . Hence, by Theorem 2, and are -equivalent, so that and are -equivalent.

Note that Theorem 3 implies that if a toroidal map admits a balanced -orientation, then admits a unique balanced -orientation that is minimal. We will exploit such orientations in our bijections.

3 Balanced -orientations on the torus

For a toroidal -angulation , we define a -orientation of as an -biorientation of such that every vertex has weight and every edge has weight .

3.1 Necessary condition on the essential girth

The following lemma gives a necessary condition for a toroidal -angulation to admit a -orientation.

Lemma 4

If a toroidal -angulation admits a -orientation then it has essential girth (i.e. it is a -toroidal map).

Proof.  Consider a toroidal -angulation endowed with a -orientation. Note that the essential girth of is at most , since faces of have degree . Let be a contractible closed walk enclosing a contractible region ; let be its length. We want to prove that .

Consider the map obtained from by keeping all the vertices and edges that are in the region , including edges and vertices of . The vertices and edges of appearing several times on the border of are duplicated so that is a -angulation of a -cycle in the plane. Let be the numbers of vertices, edges and faces of . By Euler’s formula, . All the inner faces have degree and the outer face has degree , so . Since we are considering a -orientation, the weights of the half-edges incident to inner vertices add up to and the weights of the inner half-edges add up to . Thus . Combining these three (in)equalities gives .

We will see in the Section 3.3 that, conversely, any -toroidal map admits a -orientation, and even more, it admits a balanced one.

3.2 Sufficient condition for balancedness

The next lemma shows that behaves well with respect to homotopy in -orientations:

Lemma 5

Let be a -toroidal map endowed with a -orientation, let be a non-contractible cycle of given with a traversal direction, and let be a basis for the homotopy of , such that are non-contractible cycles whose intersection is a single vertex or a common path. Let , such that is homotopic to . Then .

Proof.  Let and be the two extremities of the path (possibly , if is reduced to a single vertex). Consider a drawing of obtained by replicating a flat representation of to tile the plane. Let be a copy of in . Consider the walk starting from and following times the edges corresponding to and then times the edges corresponding to (we are going backward if is negative). This walk ends at a copy of . Since is non-contractible we have or not equal to and thus is distinct from . Let be the infinite walk obtained by replicating (forward and backward) from . Note that their might be some repetition of vertices in if the intersection of is a path. But in that case, by the choice of , the walk is almost a path, except maybe at all the transitions from “” to “”, or at all the transitions from “” to “”, where it can goes back and forth a path corresponding to the intersection of and . The existence or not of such “back and forth” parts depends on the signs of and the way are going through their common path. Figure 5 gives an example of this construction with and when intersect on a path and are oriented the same way along this path as in Figure 4.

Figure 4: Intersection of the basis.
   
   
Figure 5: Replicating “” and “” in the universal cover.

We “simplify” by removing all the parts that consist of going back and forth along a path (if any) and call the obtained walk that is now without repetition of vertices. By the choice of , the walk goes through copies of . If are no more a vertex along , because of a simplification at the transition from “” to “”, then we replace and by the next copies of along , i.e., at the transition from “” to “”.

Since is homotopic to , we can find an infinite path , that corresponds to copies of replicated, that does not intersect and situated on the right side of . Now we can find a copy of , such that lies between and without intersecting them. Choose two copies of on

such that the vectors

and are equal.

Let be the region bounded by . Let (resp. ) be the subregion of delimited by and (resp. by and ). We consider as cylinders, where the lines (or part of them) are identified. Let be the cycles of corresponding to respectively.

Let be the sum of the weights of the half-edges of incident to and in the strict interior of . Let be the sum of the weights of the half-edges of incident to and in the strict interior of . Let (resp. ) be the sum of the weights of the half-edges of incident to and in the strict interior of (resp. ). Note that corresponds to exactly one copy of , so . Similarly, (and as well) “almost” corresponds to copies of followed by copies of , except for the fact that we may have removed a back and forth part (if any). In any case we have the following:

Claim. 

Proof of the claim.  We prove the case where the common intersection of is a path (if the intersection is a single vertex, the proof is very similar and even simpler). We assume, w.l.o.g, by eventually reversing one of or , that are oriented the same way along their intersection, so we are in the situation of Figure 4.

Figure 6 shows how to compute when . Then, one can check that the weight of each half-edge of is counted exactly the same number of time positively and negatively. So everything compensates and we obtain .

Figure 6: Case .

Figure 7 shows how to compute when . As above, most of the things compensate but, in the end, we obtain equals the sum of the weights of the half-edge incident to minus the sum of the weights of the half-edge incident to . Since the sum of the weights of the half-edges at each vertex is equal to , we again conclude that .

Figure 7: Case .

One can easily be convinced that when and then the same arguments apply. The only difference is that the red or green part of the figures in the universal cover would be longer (with repetitions of and ). These parts being “smooth”, they do not affect the way we compute the equality. Finally, if one of or is equal to zero, the analysis is much simpler and the conclusion holds.

For , let be the cylinder map made of all the vertices and edges of that are in the cylinder region . Let (resp. ) be the length of (resp. ). Let be respectively the number of vertices, edges and faces of . Since is a -angulation we have . The total weight of the edges of is . Combining these equalities with Euler’s formula , one obtains . Similarly, by considering , one obtains . Thus , which gives using the claim.

Lemma 5 implies the following:

Lemma 6

Let be a -toroidal map endowed with a -orientation. If the -score of two non-contractible non-homotopic cycles of is , then the orientation is balanced.

Proof.  Consider two non-contractible non-homotopic cycles of , with a traversal direction, such that . Consider a homotopy basis of , such that are non-contractible cycles whose intersection is a single vertex or a path. Note that one can easily obtain such a basis by considering a spanning tree of , and a spanning tree of that contains no edges dual to . By Euler’s formula, there are exactly edges in that are not in nor dual to edges of . Each of these edges forms a unique cycle with . These two cycles, given with any traversal direction, form the wanted basis.

Let , such that (resp. ) is homotopic to (resp. ). Since is non-contractible we have . By possibly exchanging , we can assume, w.l.o.g., that . By Lemma 5, we have . So and thus . So or . Suppose by contradiction, that . Then , and is homotopic to . Since and are both non-contractible cycles, it is not possible that one is homotopic to a multiple of the other, with a multiple different from . So are homotopic, a contradiction. So and thus . Then by Lemma 5 we have for any non-contractible cycle of , and thus the orientation is balanced.

3.3 Existence of balanced toroidal -orientations

The main goal of this section is to prove the following existence result:

Theorem 7

Any toroidal -angulation with essential girth admits a balanced -orientation.

In the case of toroidal triangulations, essentially toroidal 3-connected maps, or essentially 4-connected toroidal triangulations, the proof of existence of analogous “balanced orientations” can be done by doing edge-contractions until reaching a map with few vertices (see [30, 12]). We do not know if such a strategy could be applied for (indeed the contraction of an edge in a -toroidal map results in some faces of size strictly less than ). So we use a different technique in the current paper.

The method consists in defining orientations that are “totally unbalanced” —which we call biased orientations— then taking a linear combinations of these biased orientations to obtain a balanced orientation but with rational weights, and finally proving that the orientation that is minimal and -equivalent to it is a balanced orientation with integer weights.

3.3.1 Biased orientations

Consider a -toroidal map , and let be a non-contractible cycle of of length given with a traversal direction. A biased orientation w.r.t.  is a -orientation of such that . Note that in a -orientation of , the sum of the weights of the half-edges incident to vertices of is and the sum of the half-edges that are on is . So we have . Thus a -orientation of is a biased orientation w.r.t.  if and only if all the half-edges incident to the left side of have weight .

The goal of this section is to prove the following lemma:

Lemma 8

Let be a -toroidal map and a non-contractible cycle of that is shortest in its homotopy class and is given with a traversal direction. Then admits a biased orientation w.r.t. .

To prove Lemma 8 we need to introduce some more general terminology concerning -orientations.

If is a subset of vertices of a graph , then denotes the set of edges of with both ends in . We need the following lemma from [7] and for which an alternative proof is suggested below:

Lemma 9 ([7])

A graph admits an -orientation if and only if , and, for every subset of vertices of , we have .

Lemma 9 can be seen as an application of Hall’s theorem regarding the existence of a perfect matching in the bipartite graph obtain from by copying times each edge , then subdividing once each edge of the resulting graph, and finally copying times each initial vertex of .

Consider a non-contractible cycle of that is a shortest cycle in its class of homotopy and given with a traversal direction. Consider the annular map obtained from by cutting along and open it as a planar map where vertices of are duplicated to form the outer face and a special inner face of . W.l.o.g., we assume that is represented such that the special inner face is on the left side of . Let be such that if is an outer-vertex of and otherwise. Let be such that if is an outer-edge of and otherwise. Then one can transform any -orientation of to a biased orientation of by gluing back the two copies of and giving to the half-edges of the weight they have on the special face of . Indeed, it is clear by the definition of and the choice of , that in the obtained -orientation of all the weights on half-edges incident to the left side of are equal to , and thus the orientation is biased w.r.t.  by the above discussion. So the existence of a biased orientation (Lemma 8), is reduced to the existence of an -orientation of . It is proved in Theorem 24 of [8] that admits an -orientation, where the proof is done first in the bipartite case (case of even ) using Lemma 9, and then the general case is derived from the bipartite case using a subdivision argument. We reproduce here in the general case the arguments given in [8] for the bipartite case, for the sake of completeness and since this is one of the key ingredients to obtain a balanced orientation of .

Lemma 10 (Theorem 24 in [8])

The annular map admits an -orientation.

Proof.  It is not difficult to check that by Euler formula that the first condition of Lemma 9 is satisfied. Let us now prove that the second condition of the lemma is also satisfied.

Let be any subset of vertices of . Suppose first that , the subgraph of induced by , is connected. We consider two cases whether contains some outer vertices of or not.

  • contains at least one outer vertex of :

    Let be the set of vertices obtained by adding to all the outer vertices of . Since equals to for outer vertices, we have . Moreover, is a subset of , so .

    Let be the number of vertices, edges and faces of . Euler’s formula says that . The outer face of has size . Since is a shortest cycle in its class of homotopy, the inner face of containing the special face of has size at least . Moreover is a -angulation, so all the other inner faces of have size at least . So finally . By combining the two (in)equalities, we obtain . So .

  • does not contain any outer vertices of :

    Let be the number of vertices, edges and faces of . Then Euler’s formula says that . The planar map has at most two faces that can be of size strictly less than : its outer face, and the face of containing the special face of . Note that these two faces are not necessarily distinct and can also be of size more than . In any case we have . By combining the two (in)equalities, we obtain . So .

In both cases, the second condition of Lemma 9 is satisfied when is connected. If is not connected, then we can sum over the different connected components to obtain the result.

By the above remarks, Lemma 10 implies Lemma 8.

3.3.2 Linear combinations of biased orientations

Consider a -toroidal map and two non-contractible non-homotopic cycles of that are both shortest cycles in their respective class of homotopy. Suppose that are given with a traversal direction. Let (resp. ) be the length of (resp. ).

Consider the four -orientations of that are biased w.r.t.  respectively. The -score of in these four orientations are given in Table 1 where are integers in and are integers in .

Table 1: -score of the orientations .

For , let be the weight function of . i.e., the function defined on the half-edges of such that the weight of a half-edge is in the -orientation . Let . Let be the weight function defined on the set of half-edges of by the following:

Note that in all cases, with weight function , the -score of both is zero. Indeed, we have:

Note also that in all cases, for the coefficient of is in , hence for every half-edge of . We denote by the sum of the coefficients, i.e.,

Note that in all cases.

Then the total -weight at any vertex (resp. edge) of equals (resp. ). Hence is the weight function of a -orientation