1.1 Some definitions
Recall that the Hausdorff distance between two non-empty subsets and of a metric space is defined as
where denotes . The Gromov-Hausdorff distance between two compact metric spaces and is defined as
where the infimum is taken over all isometric embeddings and of and into a common metric space . Note that is equal to if and only if the metric spaces and are isometric to each other. We refer the reader to e.g. [1, Section 3] for a detailed investigation of the Gromov-Hausdorff distance.
In this paper, we are considering some random graphs seen as random metric spaces and consider their convergence in distribution in the sense of the Gromov-Hausdorff distance. In general, graphs may contain loops and multiple edges. A graph is called simple if it contains no loop nor multiple edges. A graph embedded on a surface is called a map on this surface if all its faces are homeomorphic to open disks. In this paper we consider orientable surface of genus where the plane is the surface of genus , the torus the surface of genus , etc. For , a map is called a -angulation if all its faces have size . For (resp. ), such maps are respectively called triangulations (resp. quadrangulations).
1.2 Random planar maps
Let us first review some results on random planar maps. Consider a random planar map with
vertices which is uniformly distributed over a certain class of planar maps (like planar triangulations, quadrangulations or-angulations). Equip the vertex set with the graph distance . It is known that the diameter of the resulting metric space is of order (see for example  for the case of quadrangulations). Thus one can expect that the rescaled random metric spaces converge in distribution as tends to infinity toward a certain random metric space. In 2006, Schramm  suggested to use the notion of Gromov-Hausdorff distance to formalize this question by specifying the topology of this convergence. He was the first to conjecture the existence of a scaling limit for large random planar triangulations. In 2011, Le Gall  proved the existence of the scaling limit of the rescaled random metric spaces for -angulations when , or, and is even. The case solves the conjecture of Schramm. Miermont  gave an alternative proof in the case of quadrangulations . Addario-Berry and Albenque  prove the case for simple triangulations (i.e. triangulations with no loop nor multiple edges). An important aspect of all these results is that, up to a constant rescaling factor, all these classes converge toward the same object called the Brownian map.
It is natural to address the question of the existence of a scaling limit of random maps on higher genus oriented surfaces. Chapuy, Marcus and Schaeffer  extended the bijection known for planar bipartite quadrangulations to any oriented surfaces. This led Bettinelli  to show that random quadrangulations on oriented surfaces converge in distribution, at least along a subsequence. More formally:
Theorem 1 (Bettinelli ).
For and , let be a uniformly random element of the set of all angle-rooted bipartite quadrangulations with vertices on the oriented surface of genus . Then, from any increasing sequence of integers, one can extract a subsequence along which the rescaled metric spaces
converge in distribution for the Gromov-Hausdorff distance.
Contrary to the planar case, the uniqueness of the subsequential limit is not proved there. Nevertheless, a phenomenon of universality is expected: it is conjectured that the sequence does converge and that moreover, up to a deterministic multiplicative constant on the distance, the limit is the same for many models of random maps of a given genus. In genus , the conjectured limit is described in  and referred to as the toroidal Brownian map.
The present article extends Theorem 1 to the case of (essentially simple) triangulations of the torus. In that respect, it is comparable to the paper of Addario-Berry and Albenque  which did the same in the planar setup and thus our work contributes to the understanding of universality for random toroidal maps.
1.3 Main results
A contractible loop is an edge enclosing a region homeomorphic to an open disk. A pair of homotopic multiple edges is a pair of edges that have the same extremities and whose union encloses a region homeomorphic to an open disk. A graph embedded on the torus is called essentially simple if it has no contractible loop nor homotopic multiple edges. Being essentially simple for a toroidal map is the natural generalization of being simple for a planar map.
In this paper, we distinguish paths and cycles from walks and closed walks as the firsts have no repeated vertices. A triangle of a toroidal map is a closed walk of size enclosing a region that is homeomorphic to an open disk. This region is called the interior of the triangle. Note that a triangle is not necessarily a face of the map as its interior may be not empty. We say that a triangle is maximal (by inclusion) if its interior is not strictly contained in the interior of another triangle. We define the corners of a triangle as the three angles that appear in the interior of this triangle when its interior is removed (if non empty).
Our main result is the following convergence result:
For , let be a uniformly random element of the set of all essentially simple toroidal triangulations on vertices that are rooted at a corner of a maximal triangle. Then, from any increasing sequence of integers, one can extract a subsequence along which the rescaled metric spaces
converge in distribution for the Gromov-Hausdorff distance.
The reason for the particular choice of rooting in
Theorem 2 is of a technical nature due to the
bijection that we use in Section 2. It is a natural conjecture that
compactness, and thus also the existence of subsequential scaling
limits, would still hold e.g. for triangulations rooted at a
uniformly random angle. This is based on the following reasoning:
if the inside of every maximal triangle has diameter of smaller
order than , then rooting inside such a triangle rather
than at one of its corners would affect distances by a quantity that
would be smoothed out by the normalization. On the other hand,
having one maximal triangle containing vertices has very
small probability, because of the relative growths of the number of
triangulations of genus
vertices has very small probability, because of the relative growths of the number of triangulations of genusand . The remaining obstruction would be the existence of a maximal triangle with an inside containing much fewer than vertices but having diameter of order , which would presumably be ruled out by a precise control of the geometry of simple triangulations of genus . This is a possible direction for future work, but we chose not to investigate it further due to the already large size of the present paper.
We also show in an appendix that with high probability, the labeling function that we define as a crucial tool in our argument (see Section 3 for a formal definition) approximates the distance to the root up to a uniform correction (see Theorem 5
). Such a comparison estimate is an essential step in proving the uniqueness of the subsequential scaling limit, and thus the convergence, in frameworks similar to that of our main result — see for the case of genus , it is also likely that a similar argument would be applicable to quadrangulations of the torus  (those two quantities are actually equal in the case of bipartite quadrangulations on any surface with positive genus, but it seems that a bound of the order is enough).
The overall strategy for the proof of Theorem 2 is the same as in , as well as in  and : obtain a bijection between maps and simpler combinatorial objects (typically decorated trees), then show the convergence of these objects to a non-trivial continuous random limit from which relevant information can then be extracted about the original model. As a result, most of the structure of the paper is largely inspired by  (for the main argument) and  (for methods specific to triangulations).
The bijection that we use here is based on a recent generalization of Schnyder woods to higher genus [15, 14, 18]. One issue when going to higher genus is that the set of Schnyder woods of a given triangulation is no longer a single distributive lattice like in the planar case, it is rather a collection of distributive lattices. Nevertheless, it is possible to single out one of these distributive lattices, in the toroidal case, by requiring an extra property, called balanced, that defines a unique minimal element used as a canonical orientation for the toroidal triangulation. The particular properties of this canonical orientation leads to a bijection between essentially simple toroidal triangulation and particular toroidal unicellular maps  (a unicellular map is a map with only one face, i.e. the natural generalization of trees when going to higher genus). Then the main difficulty that we have to face is that the metric properties of the initial map are less apparent in the unicellular map than in the planar case or in the bipartite quadrangulations setup.
Structure of the paper
The bijection between toroidal triangulations and particular unicellular maps is presented in Section 2 with some related properties. In Section 3, we define a labeling function of the angles of a unicellular map and prove some relations with the graph distance in the corresponding triangulation. In Section 4 we explain how to decompose the particular unicellular maps given by the bijection into simpler elements with the use of Motzkin paths and well-labeled forests. In Section 5, we review some results on variants of the Brownian motion. Then the proof of Theorem 2 then proceeds in several steps. In Section 6, we study the convergence of the parameters of the discrete map in the scaling limit. In Sections 7, 8 and 9
we review and extend classical convergence results for conditioned random walks and random forests. Finally, in Section10, we combine the previous ingredients to build the proof of the main theorem. In Appendix A, we exploit the canonical orientation of the triangulation to define rightmost paths and relate them to shortest paths, thus obtaining the announced upper bound on the difference between distances and labels.
This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’avenir and the ANR project GATO (ANR-16-CE40-0009-01) funded by the French Agence National de la Recherche.
2 Bijection between toroidal triangulations and unicellular maps
For , let be the set of essentially simple toroidal triangulations on vertices that are rooted at a corner of a maximal triangle.
Consider an element of . The corner of the maximal triangle where is rooted is called the root corner. Note that, since is essentially simple, there is a unique triangle, called the root triangle, whose corner is the root corner (and this root triangle is maximal by assumption). The vertex of the root triangle corresponding to the root corner is called the root vertex. We also define, in a unique way, a particular angle of the map, called the root angle, that is the angle of that is in the interior of the root triangle, incident to the root vertex and the last one in counterclockwise order around the root vertex. Note that it is possible to retrieve the root corner from the root angle in a unique way (indeed, the root angle defines already one edge of the root triangle and the side of its interior, thus it remains to find the third vertex of the root triangle such that the interior is maximal). Thus rooting on its root corner or root angle is equivalent. We call root face, the face of containing the root angle. We introduce in the rest of this section some terminology and results adapted from  (see also ).
2.1 Toroidal unicellular maps
Recall that a unicellular map is a map with only one face. There are two types of toroidal unicellular maps since two cycles of a toroidal unicellular map may intersect either on a single vertex (square case) or on a path (hexagonal case). On the first row of Figure 1 we have represented these two cases into a square box that is often use to represent a toroidal object (its opposite sides are identified). On the second row of Figure 1 we have represented again these two cases by a square and hexagon by copying some vertices and edges of the map (here again the opposite sides are identified). Depending on what we want to look at we often move from one representation to the other in this paper. We call special the vertices of a toroidal unicellular map that are on all the cycles of the map. Thus the number of special vertices of a square (resp. hexagon) toroidal unicellular map is exactly one (resp. two).
|Square case||Hexagonal case|
Given a map, we call stem, a half-edge that is added to the map, attached to an angle of a vertex and whose other extremity is dangling in the face incident to this angle.
For , let denote the set of toroidal unicellular maps rooted on a particular angle, with exactly vertices, edges and stems distributed as follows (see figure 2 for an example in where the root angle is represented with the usual "root" symbol in the whole paper.). The vertex incident to the root angle is called the root vertex. A vertex that is not the root vertex, is incident to exactly stems if it is not a special vertex, stem if it is the special vertex of a hexagon and stem if it is the special vertex of a square. The root vertex is incident to additional stem, i.e. it is incident to exactly stems if it is not a special vertex, stems if it is the special vertex of a hexagon and stem if it is the special vertex of a square. Moreover, one of the stem incident to the root vertex, called the root stem, is incident to the root angle and just after the root angle in counterclockwise order around the root vertex.
2.2 Closure procedure
Given an element of , there is a generic way to attach step by step all the dangling extremities of the stems of to build a toroidal triangulation. Let , and, for , let be the map obtained from by attaching the extremity of a stem to an angle of the map (we explicit below which stems can be attached and how). The special face of is its only face. For , the special face of is the face on the right of the stem of that is attached to obtain (the stem is by convention oriented from its incident vertex toward its dangling part). For , the border of the special face of consists of a sequence of edges and stems. We define an admissible triple as a sequence , appearing in counterclockwise order along the border of the special face of , such that and are edges of and is a stem attached to . The closure of this admissible triple consists in attaching to , so that it creates an edge oriented from to and so that it creates a triangular face on its left side. The complete closure of consists in closing a sequence of admissible triples, i.e. for , the map is obtained from by closing any admissible triple.
Figure 3 is the hexagonal representation of the example of Figure 2 on which a complete closure is performed. We have represented here the unicellular map as an hexagon since it is easier to understand what happen in the unique face of the map. The map obtained by performing the complete closure procedure is the clique on seven vertices .
|A unicellular map of||The complete closure gives|
Note that, for , the special face of contains all the stems of . The closure of a stem reduces the number of edges on the border of the special face and the number of stems by . At the beginning, the unicellular map has edges and stems. So along the border of its special face, there are edges and stems. Thus there is exactly three more edges than stems on the border of the special face of and this is preserved while closing stems. So at each step there is necessarily at least one admissible triple and the sequence is well defined. Since the difference of three is preserved, the special face of is a quadrangle with exactly one stem. So the attachment of the last stem creates two faces that have size three and at the end is a toroidal triangulation. Note that at a given step there might be several admissible triples but their closure are independent and the order in which they are closed does not modify the obtained triangulation .
When a stem is attached on the root angle, then, by convention, the new root angle is maintained on the right side of the extremity of the stem, i.e. the root angle is maintained in the special face. A particularly important property when attaching stems is when the complete closure procedure described here never wraps over the root angle, i.e. when a stem is attached, the root angle is always on its right side in the special face. The property of never wrapping over the root angle is called safe (an analogous property is sometimes called "balanced" in the planar case but we prefer to keep the word "balanced" for something else in the current paper). Let denote the set of elements of that are safe.
Consider an element of with root angle . Then for , let be the first stem met while walking counterclockwise from in the special face of . An essential property from  is that before , at least two edges are met and thus the last two of these edges form an admissible triple with . So one can attach all the stems of by starting from the root angle and walking along the face of in counterclockwise order around this face: each time a stem is met, it is attached in order to create a triangular face on its left side. Note that in such a sequence of admissible triples closure, the last stem that is attached is the root stem of .
2.3 Canonical orientation and balanced property
For , consider an element of whose edges and stems are oriented w.r.t. the root angle as follows (see Figure 4 that corresponds to the example of Figure 2): the stems are all outgoing, and while walking clockwise around the unique face of from , the first time an edge is met, it is oriented counterclockwise w.r.t. the face of . This orientation plays a particular role and is called the canonical orientation of .
For a cycle of , given with a traversal direction, let be the number of outgoing edges and stems that are incident to the right side of minus the number of outgoing edges and stems that are incident to its left side. A unicellular map of is said to be balanced if for all its (non-contractible) cycles . Let us call the set of balanced elements of .
Figure 4 is an example of an element of . The values of the cycles of the unicellular map are much more easier to compute on the left representation.
A consequence of  (see the proof of Theorem 7 where is called and is called ), is that, for , the complete closure procedure is indeed a bijection between elements of and , that we denote in the curent paper:
Theorem 3 ().
For , there is a bijection between and .
The left of Figure 3 gives an example of a hexagonal unicellular map in . Note that on the right of Figure 3, the face containing the root angle, after the closure procedure, is indeed a maximal triangle, so the obtained triangulation is an element of if rooted on the corner of the face corresponding to the root angle.
Given an element of , the canonical orientation of , defined previously, induces an orientation of the edges of the corresponding triangulation of that is also called the canonical orientation of . Note that in this orientation of , all the vertices have outdegree exactly , we call such an orientation a -orientation. In fact this orientation corresponds to a particular -orientation that is called the minimal balanced Schnyder wood of w.r.t. to the root face (see  for more on Schnyder woods in higher genus). We extend the definition of function to by the following. For a cycle of , given with a traversal direction, let be the number of outgoing edges that are incident to the right side of minus the number of outgoing edges that are incident to its left side. As shown in , the canonical orientation of as the particular property that for all its non-contractible cycles , we call this property balanced.
Figure 5, gives the canonical orientation of obtained from the canonical orientation of its corresponding element in after a complete closure procedure.
2.4 Unrooted unicellular maps
Given an element of , we have seen that the root stem can be the last stem that is attached by the complete closure procedure. Consequently, if one removes the root stem from to obtain an unicellular map with vertices, edges and stems, one can recover the graph by applying the closure procedure on .
For , let denote the set of (non-rooted) toroidal unicellular maps, with exactly vertices, edges and stems satisfying the following: a vertex is incident to exactly stems if it is not a special vertex, stem if it is the special vertex of a hexagon and stem if it is the special vertex of a square. Thus, given an element of , the element obtained from by removing the root angle and the root stem is an element of .
Since an element of is non-rooted, it has no "canonical orientation" as define previously for elements of . Nevertheless one can still orient all the stems as outgoing and compute on the cycles of by considering only its stems in the counting (and not the edges nor the root stem anymore). For a cycle of , given with a traversal direction, let be the number of outgoing stems that are incident to the right side of minus the number of outgoing stems that are incident to its left side. A unicellular map of is said to be balanced if for all its (non-contractible) cycles . Let us call the set of elements of that are balanced.
As remarked in , an interesting property is that an element of is balanced if and only if any element of obtained from by adding a root stem anywhere in is balanced (recall that in we use the canonical orientation to compute ). Moreover, given an element of , then the element of , obtained by removing the root angle, (the canonical orientation,) and the root stem is balanced.
3 Labeling of the angles and distance properties
For , let be an element of , and the corresponding element of by Theorem 3. Let (resp. ) denotes the set of vertices (resp. edges) of . Let be the root angle of and be its root vertex. We use the same notations for the root angle and vertex of (while maintaining the root angle on the right side of every stem during the complete closure procedure, as explained in Section 2). In this section, we prove some relations between the graph distance in the triangulation and a particular labeling of the vertices defined on the unicellular map .
3.1 Definition and properties of the labeling function
Let be the number of angles of . We add a special dangling half-edge incident to the root angle of , called the root half-edge (and not considered as a stem). Let be the obtained unicellular map. We define the root angle of as the angle of just after the root half-edge in counterclockwise order around its incident vertex. Let be the sequence of consecutive angles of in clockwise order around the unique face of such that is the root angle. Note that is incident to the root half-edge. For , two angles and are either consecutive around a stem or consecutive around an edge of . We define a labeling function as follows. Let . For , let if and are consecutive around a stem, and let if they are consecutive around an edge. By definition, the unicellular map has edges and stems. While going clockwise around the unique face of , each edge is encountered twice, so . Figure 7 gives an example of the labeling function of the unicellular map of Figure 4.
Given a stem of , we define the label of as the label of the angle that is just before in counterclockwise order around its incident vertex.
The complete closure procedure is formally defined on but we can consider that it behaves on since the presence of the root half-edge in does not change the procedure as is safe (the root half-edge is maintained on the right of every stem during the closure). Let , and, for , let be the map obtained from by closing an admissible triple of . By the bijection we have that is the graph with an additional dangling half-edge incident to the root angle, we call this graph . We propagate the labeling of during the closure procedure by the following. For , when the stem of is attached, it splits an angle of into two angles of that both inherit the label of in . In other words, the complete closure procedure just splits some angles that keeps the same label on each side of the split. We still note the labeling of the angles of . It is clear that the labeling of that is obtained is independent from the order in which the admissible triples are closed. We denote the set of angles of which are splited from by the complete closure procedure. Note that for all , we have . Given a stem of , we denote the angle of corresponding to where is attached during the complete closure procedure (i.e. is attached to an angle that comes from some splittings of ).
Consider a stem of . Let , be such that is the angle just before in counterclockwise order around its incident vertex and . The fact that is safe implies that .
For , the rules that are used to define the labeling function are still valid around the special face of , i.e. the root angle of is labeled , and while walking clockwise around the special face of , the labels are increasing by one around a stem and decreasing by one along an edge until finishing at label at the last angle.
In particular, for each stem of , we have . Moreover, all the angles of that appear strictly between and in clockwise order along the unique face of have labels that are greater or equal to .
We prove the first part of the lemma by induction on . Clearly the statement is true for by definition and properties of . Suppose now that for , the statement is true for . Let be the stem of that is attached to obtained . Let be the admissible triple of involving , when is attached. Let be the angles of the special face of that appears along the admissible triple , such that appears consecutively in clockwise order around the special face. So we have that the dangling part of is attached to the angle to form . Since is safe, the root angle of is distinct from . So, by induction, the rules of the labeling function applies in from to . Thus , , . So , and the rules still apply in the special face of .
A direct consequence of the above paragraph, is that for each stem of , we have .
Suppose by contradiction that there is a stem and an angle of that appear strictly between and in clockwise order along the unique face of whose label is less or equal to . We choose such an angle whose label is minimum. With the same notations of the angles as above, since and , we have that neither nor comes from a splits of . So there exists an admissible triple , closed before is the complete closure procedure, and whose one of the two internal angles (with analogous notations as above) is (or comes from a split of ). By the rule of the labeling, we have (depending on which internal angle it is, either or ). Thus by minimality of , we have , but then , a contradiction. ∎
Consider a (non-contractible) cycle of of length that does not contain the root vertex. Then there is exactly stems attached to each side of .
As explained in Section 2.4, when one remove from the root stem, the canonical orientation and the root angle, one obtain an element of . So we have that the number of stems attached to the left and right side of are the same. In both cases, whether is a square or hexagonal unicellular map, we have that is incident to exactly stems, so there is exactly stems attached to each side of . ∎
Note that if then the conclusion of Lemma 2 is not true since there is an additional stem attached to the root vertex.
For , we have .
Assume that there exists , such that . Let . If and are consecutive along an edge, then we have . If and are separated by a stem, then, by Lemma 1, we have , so there exists such that . In both cases, there is a contradiction to the definition of . ∎
Let be the set of special vertices of (defined in Section 2). We call proper the edges and vertices of that are on at least one cycle of . Let (respectively ) be the set of proper vertices (respectively edges) of . Note that .
We call root path the (unique) shortest path of from the root vertex to a proper vertex. Note that the root path might have length if is proper. The sequence of vertices along the root path is denoted , with , and is proper. The set of edges of the root path is denoted . Let be the set of normal vertices of and be the set of normal edges of .
The canonical orientation of is the orientation of the edges and stems of that corresponds to the canonical orientation of (the root half edge added has no particular orientation). Consider an edge of with its orientation in the canonical orientation, then by the orientation rule, the angles of incident to that are on its right side have greater indices in the set than the angles that are on its left side, i.e. they are seen after while going in clockwise order around the unique face of starting from the root angle.
Consider an edge of that is oriented from to in the canonical orientation of . Let such that appear in this order in counterclockwise order around with incident to and incident to . Then we have the following (see Figure 8): and
Note first that by the labeling rule we have and . So .
Suppose first that . While going clockwise around the unique face of starting from to , we encounter only normal vertices and edges. So we go around a planar tree whose edges are encountered twice and whose number of stems is equal to twice the number of edges. This implies that and so .
The case where is quite similar. While going clockwise around the unique face of starting from to , we are in the same situation as above except that we go over the root vertex. The root vertex is incident to more stem than normal vertices and there is a jump of from the label of to around the root vertex. This implies that and so .
It only remains to consider the case where . We suppose here that is hexagonal. The case where is square can be proved similarly.
The value is equal to the number of stems minus the number of edges that are encountered while going clockwise around the unique face of starting from to , with . Each normal edge that is met is encountered twice and the number of stems that are met and attached to normal vertices is equal to exactly twice this number of edges. So there number does not affect the value . Thus we just have to look at proper edges and stems attached to proper vertices.
Let be the first special vertex that is encountered. Note that is encountered twice along the computation and the other special vertex only once. Let be the unique path of between and with no special inner vertices. Let be the length of . All the stems attached to inner vertices of are encountered exactly once and all the edges of are encountered exactly twice. Since each inner vertex of is incident to exactly two stems, and there one more edges in than inner vertices, this part results in value in the computation of .
It remains to look at the part encountered between the two copies of . This corresponds to exactly a cycle of of length , where all its edges and all the stems incident to one of its side are encountered exactly once. Note that does not belong to since . Then by Lemma 2, there are exactly stems attached to each side of . So this part results in value is the computation of .
Finally, in total we obtain and so . ∎
One can remark on Figure 8 that an incoming edge of corresponds to a variation of the labeling in counterclockwise order around its incident vertex that is always .
By Lemma 4, we can deduce the variation of the labels around the different kind of possible vertices that may appear on . They are many different such vertices, the different cases are represented on Figures 9.(a) to (). The stems are not represented on the figures, except the root stem, but their number is indicated below each figure. These stems can be incident to any angle of the figures, except the angles incident to the root half-edge that are marked with an empty set. Recall that each of this stem results in a in the variation of the labels while going counterclockwise around their incident vertex. The incoming normal edges are not represented either. There can be an arbitrary number of such edges incident to each angle of the figures. By Lemma 4, there is no variation of the labels around them. When , i.e. is the root vertex, we have represented the root stem and the root half-edge. In this particular case, there is no stem nor incoming normal edges incident to the angles incident to the root half-edge by the safe property.
|, ,||, , hexagonal|
|2 additional stems||2 additional stems||1 additional stem|
|, , square||, ,||, ,|
|0 additional stem||2 additional stems||2 additional stems|
|, , , hexagonal||, , , square||,|
|1 additional stem||0 additional stems||2 additional stems|
|, ,||, , hexagonal||, , square|
|2 additional stems||1 additional stem||0 additional stems|
For each , let be the set of angles incident to , let , and let . On Figures 9.(a) to () we have represented the position of the label and wherever the missing stems are. We also have given the value of or an inequality on it. This case analysis gives the following lemma :
For all , we have .
From Lemma 5, we obtain the following lemma.
For all , we have .
Let with extremities and . We consider two cases whether is an edge of or not.
is an edge of : While walking clockwise around the special face of from the root angle, there is an angle incident to and an angle incident to that appears consecutively. By definition of the labels, we have . Moreover by Lemma 5, we have . This implies that .
3.2 Relation with the graph distance
For , we denoted by the length (i.e. the number of edges) of a shortest path in starting at and ending at .
Given an angle of , let denote the vertex of incident to .
For all , we have .
We first prove the left inequality. Let be a shortest path in starting at and ending at , thus . We want to prove that . By Lemma 6, for all , we have . Thus we have . Moreover and . This implies that .
We now proof the right inequality. We define a walk of , starting at by the following. Let and assume that is defined for . If , then the procedure stops. If is distinct from , we consider an angle incident to such that . Let be the angle of the unique face of , just after in clockwise order around this face. If and are separated by a stem , we set . If and are consecutive along an edge of , we set . In both cases, we prove that . When and are separated by a stem , then, by Lemma 1, we have . When and are consecutive along an edge of , then, by the definition of the labeling function, we have . So, the sequence is strictly decreasing along the walk . By Lemma 3, the function is , and equal to zero only for . So the procedure ends on . Let be the length of , we have . So finally, we have . ∎
Recall that is the set of angles of and for , we have is the set of angles incident to . For , let .
For , we define the sequence of elements of by the following. Let and assume that is defined for . If , then the procedure stops. If , then we define by the following. If the two consecutive angles and of are separated by a stem , then let be such that . If and are consecutive along an edge of , then let . Note that in both cases, by Lemma 1 or the labeling rule, we have . So is decreasing by exactly one at each step. Let . Then for , we have . Thus the procedure ends on after steps, i.e. . Moreover we have that the sequence is strictly increasing since, as already remarked, by the safe property, a stem is always attached to an angle with greater index than the index of the angles incident to . We also define the corresponding walk of .
We have the following lemma:
Consider with and . Then, , and for , we have .
First, suppose by contradiction that . Then we have , so and thus . This contradicts and . So .
Let be such that . We claim that for all such that , we have . Recall that we have