1 Introduction and motivation

Asymptotic expansion of the multi-orientable random tensor model

Abstract.

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expansion in , the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.

Keywords. 3D maps, 4-regular maps, Eulerian orientations, schemes

1. Introduction and motivation

Random tensor models (see [11] for a recent review) generalize in dimension three (and higher) the celebrated matrix models (see, for example, [3] for a review). Indeed, in the same way matrix models are related to combinatorial maps [14], tensor models in dimension three are related to tensor graphs or 3D maps. A certain case of random tensor models, the so-called colored tensor models, have been intensively studied in the recent years (see [7] for a review). The graphs associated to these models are regular edge-colored graphs. An important result is the asymptotic expansion in the limit ( being the size of the tensor), an expansion which was obtained in [5]. The role played by the genus in the 2D case is played here by a distinct integer called the degree. The dominant graphs of this asymptotic expansion are the so-called melonic graphs, which correspond to particular triangulations of the three-dimensional sphere . Let us also mention here that a universality result generalizing matrix universality was obtained in the tensor case in [6].

A particularly interesting combinatorial approach for the study of these colored graphs was proposed recently by Gurău and Schaeffer in [9], where they analyze in detail the structure of colored graphs of fixed degree and perform exact and asymptotic enumeration. This analysis relies on the reduction of colored graphs to some terminal forms, called schemes. An important result proven in [9] is that the number of schemes of a given degree is finite (while the number of graphs of a given degree is infinite).

Nevertheless, a certain drawback of colored tensor models is that a large number of tensor graphs is discarded by the very definition of the model. Thus, a different type of model was initially proposed in [13], the 3D multi-orientable (MO) tensor model. This model is related to tensor graphs which correspond to 3D maps with a particular Eulerian orientation. The set of MO tensor graphs contains as a strict subset the set of colored tensor graphs (in 3D). The asymptotic expansion in the limit for the MO tensor model was studied in [2], where it was shown that the same class of tensor graphs, the melonic ones, are the dominant graphs in this limit. The sub-dominant term of this expansion was then studied in detail in [10].

In this paper we implement a Gurău-Schaeffer analysis for the MO random tensor model. We investigate in detail the general term of the asymptotic expansion in the limit . As in the colored case, this is done by defining appropriate terminal forms, the schemes. Nevertheless, our analysis is somehow more involved from a combinatorial point of view, since, as already mentioned above, a larger class of 3D maps has to be taken into consideration. Also an important difference with respect to the colored model, which only allows for integer degrees, is that the MO model allows for both half-odd-integer and integer degrees. This leads to the the fact that the dominant schemes are different from the ones identified in [9] for the colored model (interestingly, in both cases, dominant schemes are naturally associated to rooted binary trees).

Let us also mention that the analysis of this paper may further allow for the implementation of the the so-called double scaling limit for the MO tensor model. This is a particularly important mechanism for matrix models (see again [3]), making it possible to take, in a correlated way, the double limit and where is a variable counting the number of vertices of the graph and is some critical point of the generating function of schemes of a given degree.

2. Preliminaries

In this section we recall the main definitions related to MO tensor graphs.

2.1. Multi-orientable tensor graphs

A map (also called a fat-graph) is a graph (possibly with loops and multiple edges, possibly disconnected) such that at each vertex the cyclic order of the incident half-edges ( being the degree of ) in clockwise (cw) order around is specified. A corner of a map is defined as a sector between two consecutive half-edges around a vertex, so that a vertex of degree has incident corners. A 4-regular map is a map where all vertices have degree . A multi-orientable tensor graph, shortly called MO-graph hereafter, is a 4-regular map where each half-edge carries a sign, or , such that each edge has its two half-edges of opposite signs, and the two half-edges at each corner also have opposite signs. In addition, for convenience, the half-edges at each vertex are turned into 3 parallel strands, see Figure 1 for an example.

The strand in the middle is called internal, the two other ones are called external. An external strand is called left if it is on the left side of a positive half-edge or on the right side of a negative half-edge; an external strand is called right if it is on the right side of a positive half-edge or on the left side of a negative half-edge. A face of an MO-graph is a closed walk formed by a closed (cyclic) sequence of strands. External faces (faces formed by external strands) are the classical faces of the 4-regular map, while internal faces (faces formed by internal strands), also called straight faces thereafter, are not faces of the 4-regular map. Note also that external faces are either made completely of left external strands, or are made completely of right external strands; accordingly external faces are called either left or right. We finally define a rooted MO-graph as a connected MO-graph with a marked edge, which is convenient (as in the combinatorial study of maps) to avoid symmetry issues (indeed, as for maps, MO graphs are unlabelled).

2.2. The degree of an MO-graph

The degree of an MO-graph is the quantity defined by

(1)

where are respectively the numbers of connected components, vertices, and faces (including internal faces) of . For and two MO-graphs, denote by the MO-graph made of disconnected copies of and . Note that the quantities for result from adding the respective quantities from and . Hence we have:

Claim 2.1.

For two MO-graphs and , the degree of is the sum of the degrees of and . In particular, the degree of an MO-graph is the sum of the degrees of its connected components.

That follows from the following observation [13]: gives rise to three 4-regular maps (resp. , ), called the jackets of , which are obtained from by deleting straight faces (resp. deleting right faces, deleting left faces), see Figure 2 for an example. Note that is an orientable map, while and are typically only locally orientable (in a usual ribbon representation, edges have twists). Let , and be the respective genera of , , (since is orientable, , while and are in ). Let , , be the numbers of left faces, right faces, and straight faces of . Denoting by the number of edges of , the Euler relation gives , , . Since and (because the map is 4-regular) we conclude that

We call the (canonical) genus of . An MO-graph is called planar if .

Figure 1. An MO-graph, with the 3 types of strands (left strands in blue, straight strands in green, right strands in red).
Figure 2. The three jackets of the MO graph in Figure 1.

2.3. MO-graphs as oriented 4-regular maps.

Seeing edges as directed from the positive to the negative half-edge, MO-graphs may be seen as 4-regular maps where the edges are directed such that each vertex has two ingoing and two outgoing half-edges, and the two ingoing (resp. two outgoing) half-edges are opposite, see Figure 3 for an example. We call admissible orientations such orientations of 4-regular maps. This point of view of MO-graphs as oriented 4-regular maps is quite convenient and will be adopted most of the time in the rest of the article (except when we need to locally follow the strand structure). Note that the left faces are directed counterclockwise (ccw), and the right faces are directed cw; and the straight faces are closed walks that alternate in direction at each vertex they pass by.

Figure 3. The MO-graph shown in Figure 1, represented as an oriented 4-regular map.
Remark 2.1.

Since straight faces alternate at each vertex they pass by, straight faces have even length.

Note also that admissible orientations are completely characterized by the property that the faces of the 4-regular map (not including the straight faces) are either cw or ccw; this is also equivalent to coloring the faces of the map in two possible colors, say red or blue, such that at each edge there is a blue face on one side and a red face on the other side (this is also directly visible given the definition with strands, where right faces are red and left faces are blue, see Figure 1). In other words, a 4-regular map can be endowed with an admissible orientation iff the dual of every of its connected components is a bipartite quadrangulation. If this is the case, and if there is a marked directed edge in each connected component, then there is a unique admissible orientation that fits with the prescribed edge directions (existence follows from the above discussion, and there is clearly a unique way to propagate the directions starting from the marked directed edges).

Recall that, in the context of maps, a rooted map means a connected map with a marked directed edge. The above paragraph yields the following statement, which we think is worth mentioning even if it will not be needed to perform the combinatorial study (scheme extraction and associated analysis) of MO-graphs:

Remark 2.2.

Rooted MO-graphs having genus and vertices may be identified with rooted 4-regular maps having genus and vertices, and with the property that the dual rooted quadrangulation is bipartite (these are themselves well-known to be in bijection with rooted connected maps having genus and edges). In genus (planar case), all rooted 4-regular maps have this property.

We conclude this subsection with a simple lemma relating the degree and the genus to the number of straight faces:

Lemma 2.1.

Let be a connected MO-graph. Let be the number of vertices, the number of straight faces, and for each , the number of straight faces of length in . Then the quantity satisfies

where and are respectively the degree and the genus of . Hence .

Proof.

Since the number of edges is twice the number of vertices, we have , so that , hence . Let , be the numbers of left faces and right faces of . We have seen in Section 2.2 that , where and

Substracting the sum of the two last equalities from the first one, we obtain:

so that . ∎

Remark 2.3.

For , according to Remark 2.2, rooted MO-graphs correspond to rooted 4-regular planar maps. Seeing such a map as the planar projection of an entangled link that lives in the 3D space (vertices of the map correspond to crossings, where the under/over information is omitted), is classically interpreted as the number of knot-components. Lemma 2.1 gives . Since and , the extremal cases are: (1) , in which case ; (2) , in which case (the MO-graph of Figure 3, having , , and thus , is intermediate). While the case is combinatorially well-understood (as will be recalled in Section 3.1), the case is notoriously difficult: in that case, 4-regular planar maps are projected diagrams of one-component links (i.e., knot diagrams), see [12] and references therein.

We will go back to this remark in Section 7 (on generating functions), where we will establish, for each fixed , the asymptotic enumeration of rooted 4-regular planar maps with vertices and with knot-components, as goes to infinity.

2.4. Regular colored graphs for as a subfamily of MO-graphs

As recalled here from [13], MO-graphs form a superfamily of the well studied regular colored graphs of dimension . For a regular colored graph of dimension is defined as a -regular bipartite graph (vertices are either black or white) where the edges have a color in , such that at each vertex the incident edges have different colors. A rooted colored graph is a connected colored graph with a marked edge of color . For , a face of type in a colored graph of dimension is a cycle made of edges that alternate colors in ; let be the number of faces of type in . Let be the number of connected components, the number of vertices and the total number of faces of , i.e., . The degree of is the integer given by

(2)
Figure 4. Left: a rooted colored graph in dimensions. Right: the induced rooted MO-graph.

Note that a colored graph has a canonical realization as a map, where the edge colors in cw (resp. ccw) order around black (resp. white) vertices are . Let be a rooted colored graph of dimension that is canonically embedded. Orienting the edges of even color from the black to the white extremity and the edges of odd color from the white to the black extremity, we obtain a rooted MO-graph , see Figure 4 for an example. Clearly this gives an injective mapping, since there is a unique way (when possible) to propagate the edge colors starting from the root-edge. Hence, rooted colored graphs of dimension form a subfamily of rooted MO-graphs. In addition

so that and have the same total number of faces. Hence the degree formula for colored graphs is consistent with the degree formula for MO-graphs.

3. From MO-graphs to schemes

3.1. Extracting the melon-free core of a rooted MO-graph

Figure 5. Left: a melon. Right: a triple edge that does not form a melon.

From now on, it is convenient to consider that a “fake” vertex of degree , called the root-vertex, is inserted in the middle of the root-edge of any rooted MO-graph (so that the root-edge is turned into two edges). By convention also, it is convenient to introduce the following MO-graph: the cycle-graph is defined as an oriented self-loop carrying no vertex. The cycle-graph is connected, has , (one face in each type), hence has degree . In its rooted version, the (rooted) cycle-graph is made of an oriented loop incident to the root-vertex. In a (possibly rooted) MO-graph , a melon is a triple edge such that none of the edges is the root-edge (if is rooted), form a left face of length , form a right face of length , and form a straight face of length , see Figure 5. Define the removal of a melon as the operation below (where possibly , and possibly or might be the root if the MO-graph is rooted):

The reverse operation (where is allowed to be a loop, and is allowed to be incident to the root-vertex if the MO-graph is rooted) is called the insertion of a melon at an edge.

Figure 6. A rooted melonic graph is one that can be built from the cycle-graph, by successive insertions of melons at edges.

A rooted MO-graph is called melonic if it can be reduced to the rooted cycle-graph by successive removals of melons, see Figure 6 for an example. An important remark is that, for a rooted melonic MO-graph, any greedy sequence of melon removals terminates at the cycle-graph. It is known [2] that rooted MO-graphs of degree are exactly the rooted melonic graphs. For a (possibly rooted) MO-graph, an edge of (possibly a loop, possibly incident to the root-vertex), and a rooted MO-graph, define the operation of substituting by in as in the following generic drawing:

We have the following bijective statement, which is the counterpart for MO-graphs of [9, Theo. 4]:

Proposition 3.1.

Each rooted MO-graph is uniquely obtained as a rooted melon-free MO-graph —called the melon-free core of — where each edge is substituted by a rooted melonic MO-graph (possibly the rooted cycle-graph, in which case is unchanged). In addition, the degree of equals the degree of .

Proof.

Let be the rooted melon-free graph obtained from after performing a maximal greedy sequence of melon removals. Conversely, is obtained from where a sequence of melon insertions is performed. Hence is equal to where each edge is substituted by a rooted melonic graph . This gives the existence of a melon-free core. Uniqueness of the melon-free core is given by the observation that any other maximal greedy sequence of melon removals starting from has to progressively shell the melonic components , hence terminates at . Finally, it is clear that and have the same degree, since a melon insertion preserves the degree (it preserves the number of connected components, increases the number of vertices by , and increases the number of faces by ). ∎

Figure 7. Left: the three types of dipoles. Right: two examples of double edges that do not form a dipole.
Figure 8. Left: the clockwise infinity graph. Right: the counterclockwise infinity graph. Both have two faces of length , two faces of length , and have degree .

3.2. Extracting the scheme of a rooted melon-free MO-graph

In this section we go a step further and define the process of extracting a scheme from a rooted melon-free MO-graph.

Define a 2-dipole, shortly a dipole, in a (possibly rooted) MO-graph as a face of length incident to two distinct vertices, and not passing by the root if is rooted. Accordingly, one has three distinct types of dipoles: L, R, or S, see Figure 7 (left part). As the figure shows, each dipole has two exterior half-edges on one side and two exterior half-edges on the other side. Notice also that a double edge does not necessarily delimit a dipole, as shown in Figure 7 (right part).

Remark 3.1.

A face of length is actually always incident to two distinct vertices, except in the MO-graphs that are made of one vertex and two loops. These two graphs, shown in Figure 8, are called the clockwise and the counterclockwise infinity graph, respectively. They have no dipole but have two faces of length .

In an MO-graph , define a chain as a sequence of dipoles (not passing by the root if is rooted) such that for each , and are connected by two edges involving two half-edges on the same side of and two half-edges on the same side of , see Figure 9. A chain is called unbroken if all the dipoles are of the same type. A broken chain is a chain which is not unbroken. A proper chain is a chain of at least two dipoles. A proper chain is called maximal if it cannot be extended into a larger proper chain. By very similar arguments as in Lemma 8 of [9] one obtains the following result:

Figure 9. Left: a chain of dipoles. Right: a sequence of 3 dipoles that do not form a chain (the 3rd element, a dipole of type , is connected to the 2nd element by two half-edges on opposite sides).
Claim 3.1.

In a rooted MO-graph, any two maximal proper chains are vertex-disjoint.

Let be a rooted melon-free MO-graph. The scheme of is the graph obtained by simultaneously replacing any maximal proper chain of by a so-called chain-vertex, as shown in Figure 12, see also Figure 10 for an example.

Figure 10. Left: a rooted melon-free MO-graph. Right: the associated scheme.
Figure 11. The two types of vertices (standard vertex or chain-vertex, which can be labelled by either ) in an MO-graph with chain-vertices.
Figure 12. Top-part: examples of the 4 types of unbroken proper chains; Bottom-part: example of a broken proper chain. In each case, the graphical symbol for the chain-vertex (in the associated scheme) is shown.
Figure 13. The configurations of the strands for each type of chain-vertex.

Since a scheme is not an MO-graph (due to the presence of chain-vertices), we need to extend the definition and main properties seen so far in order to allow for chain-vertices. In the following, an MO-graph with chain-vertices is an oriented 4-regular map with two types of vertices: standard vertices and chain-vertices (each chain-vertex being labelled by either ), so as to satisfy the local rules of Figure 11. Such a graph is possibly rooted, i.e., has a fake-vertex of degree in the middle of some edge. Note that the class of (rooted) MO-graph with chain-vertices is larger than the class of schemes (we will see below which rooted MO-graphs with chain-vertices are schemes). In order to compute the number of faces of an MO-graph with chain-vertices, we need to specify locally the strand structure at every type of chain-vertex; as for MO-graphs, we imagine here that each edge is turned into parallel strands and we have to specify at each chain-vertex how the incident strands go through the vertex. The specification is given by Figure 13; in the case of an unbroken chain-vertex the two strands that go through the chain-vertex are called the crossing strands at that chain-vertex. Note that this specification gives the natural strand structure to expect whenever a chain-vertex is to be consistently substituted by a chain (for instance a chain-vertex of type is to be susbtituted by an unbroken chain of dipoles of type ), that is, the strand structure of reflects how the strands arriving at are routed (some bouncing back, some going through ). Then, as for classical MO-graphs, a face is a closed walk formed from strands. The degree of is defined as

(3)

where , , are as usual the numbers of connected components, standard vertices, and faces, and where and stand respectively for the numbers of unbroken chain-vertices and broken chain-vertices.

An edge of is said to be adjacent to a chain-vertex if the two half-edges of are the two half-edges on the same side of a chain-vertex of . Then is said to be melon-free if it has no melon nor an edge adjacent to a chain-vertex. It is easy to see that an MO-graph is melon-free iff its scheme is melon-free.

In a melon-free (possibly rooted) MO-graph with chain-vertices, define a chain as a sequence of elements that are either dipoles (not passing by the root if is rooted) or chain-vertices, such that for each , and are connected by two edges involving two half-edges on the same side of and two half-edges on the same side of , see Figure 14. A proper chain is a chain of at least two elements.

Figure 14. A chain of five elements in an MO-graph with chain-vertices: it consists of two dipoles and three chain-vertices.

Now define a reduced scheme as a rooted melon-free MO-graph with chain-vertices and with no proper chain. By construction, the scheme of a rooted melon-free MO-graph (with no chain-vertices) is a reduced scheme. Claim 3.1 then easily yields the following bijective statement:

Proposition 3.2.

Every rooted melon-free MO-graph is uniquely obtained as a reduced scheme where each chain-vertex is consistently substituted by a chain of at least two dipoles (consistent means that if the chain-vertex is of type , then the substituted chain is an unbroken chain of L-dipoles, etc).

The following result ensures that the degree definition for MO-graphs with chain-vertices is consistent with the replacement of chains by chain-vertices:

Lemma 3.1.

Let be an MO-graph with chain-vertices. And let be an MO-graph with chain-vertices obtained from by consistently substituting a chain-vertex by a chain of dipoles. Then the degrees of and are the same.

Proof.

Let be the chain-vertex where the substitution takes place, and let be the number of dipoles in the substituted chain. Denote by the parameters (degree, numbers of connected components, standard vertices, faces, unbroken chain-vertices, broken chain-vertices) for , and by the parameters for . Clearly , . If is unbroken, then and . Say is of type (the arguments for types are similar). Then the strand structure remains the same, except for new faces of type (of degree ), new faces of type (of degree ) and new faces of type (of degree ) inside the substituted chain. Hence . Consequently, . If is broken, then and . The strand structure remains the same, except for new faces of each type inside the substituted chain (an easy case inspection ensures that substituting a chain of two dipoles of different types always bring new faces, one of each type, and any additional dipole brings new faces, one of each type). Hence . Consequently, . ∎

Corollary 3.1.

A rooted melon-free MO-graph has the same degree as its scheme.

4. Finiteness of the set of reduced schemes at fixed degree

The main result in this section is the following:

Proposition 4.1.

For each , the set of reduced schemes of degree is finite.

Similarly as in [9], this result will be obtained from two successive lemmas, proved respectively in Section 4.2 and Section 4.3:

Lemma 4.1.

For each reduced scheme of degree , the sum of the numbers of dipoles and chain-vertices satisfies .

Lemma 4.2.

For and , there is a constant (depending only on and ) such that any connected unrooted MO-graph (without chain-vertices) of degree with at most dipoles has at most vertices.

Figure 15. The drawings show a choice of canonical substitution of chain-vertices by proper chains that preserves the strand structure (hence the degree) and yields an injective mapping from MO-graphs with chain-vertices to MO-graphs without chain-vertices.

Let us show how Lemmas 4.1 and 4.2 imply Proposition 4.1. Let be a reduced scheme of degree , and let be the rooted MO-graph with no chain-vertices obtained by substituting each chain-vertex of by a chain of dipoles of length or as shown in Figure 15. As already seen in Lemma 3.1, such substitutions preserve the degree, and the mapping is injective. In addition, the number of dipoles of is at most three times the total number of dipoles and chain-vertices of , hence has at most dipoles according to Lemma 4.1. Unrooting might increase the number of dipoles by up to (recall that dipoles are not counted in rooted MO-graphs if they pass by the root, and at most dipoles can pass by the root, the case of additional dipoles happening when the root-edge belongs to a melon). Hence, writing , Lemma 4.2 ensures that has at most vertices. Since there is an injective mapping from reduced schemes of degree to rooted MO-graphs of size bounded by the fixed quantity , we conclude that the number of reduced schemes of degree is finite.

We finally state the following “loop-removal” lemma, which will be useful at some points in the following:

Figure 16. Removing a loop in an MO-graph.
Lemma 4.3.

Let be an MO-graph of degree , let be a loop of , and let be the MO-graph obtained from by erasing the loop and its incident vertex, as shown in Figure 16. Then has degree . Hence has at most loops.

Proof.

Clearly has the same number of connected components as , has one vertex less, and has one face less (which has length ). ∎

4.1. Analysis of the removal of dipoles and of chain-vertices

We analyze here how the degree of an MO-graph with chain-vertices (not necessarily a reduced scheme, possibly with melons) evolves when removing a chain-vertex. We then have a similar analysis for the removal of a dipole.

Figure 17. The first drawing shows the removal of a chain-vertex (whatever its type). The next drawings show the removal of a dipole in each of the three types (L,R,S).

Let be an MO-graph with chain-vertices, and let be a chain-vertex of . The removal of consists of the following operations (see the left part of Figure 17):

  1. delete from ;

  2. on each side of , connect together the two detached legs (without creating a new vertex).

Denote by the resulting graph. The chain-vertex is said to be non-separating if has the same number of connected components as , and separating otherwise (in which case has one more connected component). In the separating case, it might be that one of the two resulting connected components is the cycle-graph (this happens if two half-edges on one side of belong to the same edge). Let be the parameters for (degree, numbers of connected components, standard vertices, faces, unbroken chain-vertices, broken chain-vertices), and let be the parameters for .

Assume first that is a broken chain-vertex. We clearly have , (the strand structure is the same before and after removal), , and (resp. ) if is non-separating (resp. separating). Hence, if is separating, then , so that ; and if is non-separating, then , so that .

Assume now that is an unbroken chain-vertex. We clearly have , , , and (resp. ) if is non-separating (resp. separating). If the two crossing strands at belong to a same face, then that face splits into two faces when removing , hence ; as opposed to that, if the two crossing strands belong to different faces, then these two faces are merged when removing , hence . Moreover, the two crossing strands are easily seen to always be in the same face if is separating. We conclude that, if is separating, then , so that ; and if is non-separating then , where , hence either or .

We now consider the operation of removing a dipole from , removal which consists of the following operations (the 3 cases are shown in the right-part of Figure 17):

  1. delete the two vertices and the two edges of from ;

  2. on each side of , connect together the two detached legs (without creating a new vertex).

Let be the resulting graph. Again, is called non-separating if has as many connected components as , and separating otherwise. Observe that the operation of removing is the same as substituting by an unbroken chain-vertex (of the same type as , so that the substitution does not change the degree according to Lemma 3.1) and then removing . From the discussion on the degree variation when removing , we conclude that, if is separating, then the degree is the same in as in , and if is non-separating then the degree of equals the degree of minus a quantity that is either or .

This analysis thus leads to the following statement:

Lemma 4.4.

The degree is unchanged when removing a separating chain-vertex or a separating dipole (hence, by Claim 2.1, the degree is distributed among the resulting components). The degree decreases by when removing a non-separating broken chain-vertex. The degree decreases by or when removing a non-separating unbroken chain-vertex or a non-separating dipole.

Corollary 4.1.

An MO-graph with chain-vertices and with degree has no non-separating dipole nor non-separating chain-vertex.

4.2. Proof of Lemma 4.1

We show Lemma 4.1, i.e., that for each reduced scheme of degree , the sum of the numbers of dipoles and chain-vertices in satisfies .

Figure 18. In the worst case, removing a dipole might decrease by the total number of (uncolored) dipoles.
Figure 19. The first drawing shows a reduced scheme . Then, iteratively, one removes at each step a non-separating dipole or a non-separating chain-vertex (at each step the non-separating dipole or chain-vertex to be removed next is surrounded). Let be the MO-graph with chain-vertices thus obtained (where colored edges are drawn bolder). As the last two drawings show, the removal of uncolored dipoles and chain-vertices (which are all separating) of yields a tree of components (a tree edge is labelled if it comes from a chain-vertex of type and is labelled if it comes from an uncolored dipole of type ).

As a first step, starting from , as long as there is at least one non-separating dipole or one non-separating chain-vertex, delete it (as in Figure 17) and color the two resulting edges. Let be the resulting graph with chain-vertices, and let be the number of removals from to . According to Lemma 4.4, each non-separating removal decreases the degree by at least , hence , has degree at most , and has at most blue edges. During the sequence of steps from to , denote by the current sum of the numbers of chain-vertices and uncolored dipoles (dipoles with no colored edge). It is easy to see that each removal decreases by at most (the worst case is shown in Figure 18). Hence, if we denote by the value of for , we have .

Call a connection a chain-vertex or uncolored dipole of . Since all dipoles and chain-vertices of are separating, can be seen as a tree of “components” (see the last row of Figure 19 for an example), where the edges of correspond to the connections of , and the nodes of correspond to the connected components of left after removing the connections (note that some of these connected components might be cycle-graphs, including the component carrying the root). For a component of , the adjacency order of is the number of adjacent components in . An edge of resulting from a (removed) connection is called a marked edge. As shown in Figure 20, every marked edge is involved in connections (hence the adjacency order of a component is at least its number of marked edges), and results from the merger of edges of ; is called colored if at least one of these edges is colored. A component is called colored if if it bears either a colored edge of left untouched by removals, or a colored marked edge; and it is called uncolored otherwise. Thus, there are at most colored components in .

Figure 20. For each marked edge of a component of such that is involved in connections, results from the merger of edges of ( in the example, results from the merger of ).

Similarly as in [9], an important remark is the following:

Claim 4.1.

An uncolored component of of degree and not containing the root has adjacency order at least .

Proof.

The first unrooted MO-graphs of degree are the cycle-graph (no vertex) and the so-called “quadruple-edge graph” (the MO-graph whose underlying map is of genus with two vertices connected by edges). And any other MO-graph of degree can be obtained from the quadruple-edge graph by successive melon insertions. This easily implies that, if is not the cycle-graph and has adjacency order smaller than (hence has less than marked edges), then has a dipole not passing by any marked edge, a contradiction. If is the cycle-graph and has adjacency order , then it clearly yields a melon in , a contradiction. Finally, if is the cycle-graph and has adjacency order , then the two edges of at —each of which either corresponds to a separating dipole or to a chain-vertex of — together form a proper chain (of two elements) of , a contradiction. ∎

Denote by the number of uncolored components in of degree and not containing the root, the number of other components of degree (colored or containing the root), and the number of components of positive degree. The degrees of the components add up to the degree of (recall that, for separating removals, the degree is distributed among the components), and each component of positive degree has degree at least , hence . The components counted by have at least neighbours in , hence . Since has at most colored components, we have (the accounting for the component containing the root). And since is a tree, we have

hence, if we eliminate using , we obtain

so that . Since , we conclude that .

4.3. Proof of Lemma 4.2

We prove here Lemma 4.2, i.e., that for any and , there exists some constant such that any connected unrooted MO-graph (with no chain-vertices) of degree with at most dipoles has at most vertices, so that there are finitely many MO-graphs of degree with dipoles. The bound we obtain is linear in and , with quite large multiplicative constants (we have not pushed to improve the bound, also to avoid making the proof too complicated).

The proof is a bit more technical than the analogous statement for colored graphs in [9], because in the MO model, only straight faces have even length, while left and right faces are allowed to have odd lengths. Precisely, our proof strategy is to show in a series of claims that some properties (e.g. being incident to a dipole) are satisfied only by a limited (bounded by a quantity depending only on and ) number of vertices. Hence if could be arbitrarily large, it would have vertices not safisfying these properties. As we will see, this would yield the contradiction that has to be reduced to a certain MO-graph with vertices.

In all the proof, denotes a connected unrooted MO graph with no chain-vertices, of degree and with at most dipoles; are respectively the numbers of vertices, edges, and faces of . Moreover, for , is the number of straight faces of length . According to Remark 3.1, the only MO-graphs where the set of dipoles is not equal to the set of faces of length are the two infinity graphs (both having one vertex). Hence we can assume that is not equal to these two graphs. Thus , which gives:

Claim 4.2.

There are at most vertices of that are incident to a dipole.

Also, Lemma 2.1 directly implies

Since for , the total length of straight faces of length larger than is bounded by , so that we obtain:

Claim 4.3.

There are at most vertices of incident to a straight face of length larger than .

A cycle is called self-intersecting if it passes twice by a same vertex.

Figure 21. On the left the unique map configuration for a self-intersecting straight face of length that does not create a loop. On the right we see that it is not possible to orient the edges so as to satisfy the local condition at the vertices and if the local condition is satisfied at .
Claim 4.4.

There are at most self-intersecting straight faces of length in . Hence there are at most vertices incident to such faces.

Proof.

As we see in Figure 21, a self-intersecting straight face of length must contain a loop, which yields the first statement according to Lemma 4.3 (there are at most loops). The second statement just follows from the fact that these cycles (due to the self-intersection) have at most vertices. ∎

Given a non-self-intersecting cycle of , a chordal path for is a path that starts from , ends at (possibly at the same vertex), and is outside of inbetween. In the context of maps, we call a chordal path faulty if it starts on one side of and ends on the other side of . A cycle is called faulty if it has a faulty chordal path, and, for , is called -faulty if it has a chordal path of length at most . For a faulty cycle, we denote by the union of and of a (canonically chosen) shortest possible faulty chordal path of ; is called the faulty extension of .

Claim 4.5.

Let . There are at most 4-faulty cycles of length in . Hence there are at most vertices incident to such cycles.

Proof.

For a faulty cycle, is clearly a topological minor of of genus . Hence, if we have vertex-disjoint faulty extensions of faulty cycles, these yield a topological minor of genus . Since the genus of is bounded by , we conclude that there can not be more than vertex-disjoint faulty extensions of faulty cycles. Let be the set of 4-faulty cycles of length of , and let (where we distinguish the cycle from the faulty path in every faulty extension, so that ). Any has at most vertices, and an easy calculation ensures that any given vertex of belongs to at most elements of (this can probably be improved, but we just aim at a certain fixed bound). Hence can intersect at most other elements of . We easily conclude that (otherwise one could construct incrementally a subset of vertex-disjoint elements of ). ∎

Let be the set of vertices of that are either incident to a dipole, or incident to a straight face of length larger than , or incident to a self-intersecting straight face of length , or incident to a 4-faulty cycle of length . The four claims above give

(4)
Figure 22. The situation in the proof of Lemma 4.2

And let be the set of vertices in or adjacent to a vertex in . Since is -regular, . We are now going to show that if has a vertex not in , then is actually reduced to having only vertices. Since , it is incident to two non-intersecting straight faces and , both of length . Since is not faulty, it has to meet at a vertex different from (otherwise would be a faulty chordal path for ). Let be an intersection of and different from . For , let be the distance of from on (note that ). If then either the two corresponding edges of and form an external dipole, or otherwise is a faulty chordal edge for . Both cases are excluded since , hence . We write the cycle as and the cycle as , see Figure 22. Let be the straight face (non-intersecting and of length , since ) passing by and different from . Since , is not faulty, hence it intersects the 4-cycle at a vertex different from ; this other intersection must be at because and have their incident edges already in . Similarly intersects the 4-cycle at a vertex different from , and this other intersection must be ; and intersects the 4-cycle at a vertex different from , and this other intersection must be , see Figure 22. Hence forms a connected induced subgraph that saturates all its vertices (each of the vertices of is incident to edges of ). Hence . In other words, if has a vertex outside of , then has vertices.

We can now easily conclude the proof. Define , and assume has more than vertices. Since , has a vertex not in , which implies that is reduced to a graph with vertices, giving a contradiction.

5. MO-graphs of degree

In this section we recover the results of [10] about the structure of MO-graphs of degree , using ingredients from the preceding sections.

Proposition 5.1.

The only unrooted connected melon-free MO-graphs of degree are the cw and the ccw infinity graph.

Proof.

Assume there is a connected melon-free MO-graph of degree different from the cw or the ccw infinity graph. According to Remark 3.1 the set of dipoles of equals the set of its faces of length . Note that in degree , the genus has to be (since is a nonnegative integer bounded by ). Hence Lemma 2.1 gives , so that , i.e., has a dipole (of type ). Since , has to be separating according to Corollary 4.1. Let and be the connected components resulting from the removal of . Since is separating, the degrees of and add up to , hence one has degree and the other has degree . By convention let be the one of degree , considered as rooted at the edge resulting from the deletion of . There are two cases: (i) if is the cycle-graph, then is part of a melon of , giving a contradiction; (ii) if is not the cycle-graph, then it has a melon (since rooted MO-graphs of degree are melonic), giving again a contradiction. ∎

This yields the following bijective result, where we recover [10, Theo. 3.1]:

Corollary 5.1.

Every unrooted connected MO-graph of degree is uniquely obtained from the cw or the ccw infinity graph where each of the two edges is substituted by a rooted melonic graph.

Proof.

All the arguments in the proof of Proposition 3.1 (extraction of a unique melon-free core) can be directly recycled to give the following statement: “for , every unrooted connected MO-graph of degree is uniquely obtained as an unrooted connected melon-free MO-graph of degree where each edge is substituted by a rooted melonic graph”, which —together with Proposition 5.1— yields the result. ∎

Another direct consequence of Proposition 5.1 is the following statement that will prove useful in the next section:

Corollary 5.2.

Every rooted connected MO-graph of degree different from the (rooted) cw or ccw infinity graph has a dipole.

Proof.

According to Proposition 5.1, the underlying unrooted MO-graph (obtained by erasing the fake-vertex of degree ) has a melon, hence there is a dipole not including the root edge; by definition this dipole is also a dipole of the rooted MO-graph (recall that dipoles of rooted MO-graphs are required not to pass by the root). ∎

6. Dominant schemes

A reduced scheme of degree is called dominant if it maximizes (over reduced schemes of degree ) the number of broken chain-vertices (as we will see in Section 7, determines the singularity order of the generating function of rooted MO-graphs of degree and reduced scheme , the larger the larger the singularity order). In this section we show that, in each degree , the maximal number of broken chain-vertices is , and we precisely determine what are the dominant schemes.

Let be a reduced scheme of degree , and let be the number of broken chain-vertices of . We now determine a bound for in terms of . The discussion is very similar to the one of Section 4.2, with the difference that only the broken chain-vertices are removed. First, as long as there is a non-separating broken chain-vertex, we remove it (as shown in Figure 17) and color the two resulting edges. Let be the number of such removals until all broken chain-vertices are separating; at this stage, denote by the resulting MO-graph with chain-vertices, which has degree according to Lemma 4.4, and has at most colored edges. Then can be seen as a tree of components that are obtained by removing all separating broken chain-vertices. It is also convenient to consider the tree of components that are obtained by removing all separating chain-vertices and all separating uncolored dipoles —dipoles with no colored edge— of (in and some components of degree might be cycle-graphs). Note that is a refinement of , i.e., each component of “occupies” a subtree of . For a component of (resp. of ), the adjacency order of is defined as the number of adjacent components in (resp. in ). An edge of resulting from a separating removal for (resp. ) is called a marked edge. Such an edge results from the merger of edges of . Then is called colored if at least one of these edges is a colored edge of . And a component of (resp. of ) is called uncolored if it bears no colored edge (marked or unmarked). Since has at most colored edges, (resp. ) has at most colored components.

Consider an uncolored component of of degree and not containing the root. Note that has no non-separating dipoles nor non-separating chain-vertices, by Corollary 4.1. In addition, by the arguments of Claim 4.1, any component of must have adjacency order at least , otherwise it would yield a melon or a proper chain in . This clearly implies that can not have adjacency order smaller than , and if it has adjacency order , then consists of a unique component.

Let be the number of components of of degree , uncolored, and not containing the root. Let be the number of other components of degree . Let be the number of components of positive degree. Moreover, let be the number of edges of , which is also the number of broken chain-vertices of , so that .

The degrees of the components add up to the degree of (recall that, for separating removals, the degree is distributed among the components), and each component of positive degree has degree at least , hence

The components counted by have at least neighbours in , hence

The graph has at most blue edges, hence

where the accounts for the component containing the root. Since is a tree, we have

hence, eliminating using , we obtain

so that . Hence .

If a scheme is such that reaches the upper bound , then it implies that all the above inequalities are tight, hence (all broken chain-vertices are separating), (the component containing the root has degree ), (all the components of positive degree and the component containing the root are leaves of , and the other components of degree have neighbours in ), and (all positive degree components have degree ).

We take a closer look at the components of degree . Let be such a component not containing the root; note that has no non-separating dipole nor non-separating chain-vertex according to Corollary 4.1. Since has adjacency order and is uncolored (because ), as already mentioned must have a single component, hence has no chain-vertex and any dipole of must pass by a marked edge of . Recall that an unrooted MO-graph of degree is either the cycle-graph, or can be obtained from the “quadruple-edge” MO-graph (two vertices connected by edges) by successive insertions of melons. Note that, as soon as at least one melon is inserted in the quadruple-edge MO-graph, there are two vertex-disjoint melons. Hence, an MO-graph of degree with at most marked edges and strictly more than vertices has a dipole that avoids the marked edges. It follows that has to be either the cycle-graph or the quadruple-edge graph. Now, if is the component containing the root, since it has only one adjacent component , it has to be the (rooted) cycle-graph. Indeed, the case of a rooted quadruple-edge MO-graph is excluded; in that case, among the 4 edges of , one carries the root, one is marked, and the two other ones form a dipole that, together with the chain-vertex connecting to , would form a proper chain of , a contradiction.

We now take a closer look at components of positive degrees (of degree ). Let be such a component (which has a unique marked edge since it is at a non-root leaf of ). Since has degree it can not have non-separating chain-vertices or non-separating unbroken chain-vertices, according to Corollary 4.1. And by the arguments of Claim 4.1, components of degree of must have adjacency order at least . Since the degrees of components in add up to , this easily implies that the unique possibility is having a single node. Hence has no dipole avoiding the marked edge nor chain-vertices, so that must be the cw or the ccw infinity graph, according to Corollary 5.2.

Figure 23. (a) A rooted binary tree with leaves. (b) One of the dominant schemes of degree arising from .
Proposition 6.1.

For , the dominant schemes of degree arise from rooted binary trees (see Figure 23 for an example) with leaves, inner nodes, and edges, where the root-leaf is occupied by the rooted cycle-graph, the other leaves are occupied by (cw or ccw) infinity graphs, the inner nodes are occupied either by the cycle-graph or by the quadruple-edge graph, and the edges are occupied by separating broken chain-vertices.

Each rooted binary tree with leaves yields dominant schemes.

Proof.

In the analysis above we have seen that and that any scheme with must be of the stated form. Conversely any scheme of the stated form is a valid scheme (no melon nor proper chain) of degree with chain-vertices. Hence these schemes are exactly the dominant schemes at degree . Each rooted binary tree with leaves gives rise to dominant schemes. Indeed, at each inner node, one has to decide whether it is occupied by the cycle-graph or by the quadruple-edge graph, and in the second case, one has to choose among the three possible configurations for the free edge (the unique edge not involved in any connection) since the straight face passing by the free edge can go either toward the left child, or the right child, or the parent. Finally, one has to decide at each non-root leaf if it is occupied by the cw or the ccw infinity graph. ∎

Remark 6.1.

As we have recalled in Section 2.4, regular colored graphs in dimension naturally form a subfamily of MO-graphs. However, since half-integer degrees are not possible in the model of regular colored graphs, the dominant schemes differ; as shown in [9], in degree , the dominant schemes are associated to rooted binary trees with leaves (and inner nodes), where the root-leaf is occupied by a root-melon, while the non-root leaves are occupied by the unique scheme of degree .

7. Generating functions and asymptotic enumeration

Let , and let be the (finite) set of reduced schemes of degree . For each , let be the generating function of rooted melon-free MO-graphs of reduced scheme , where marks half the number of non-root vertices (i.e., for , a weight is given to rooted MO graphs with non-root vertices). Let be half the number of non-root standard vertices of , the number of broken chain-vertices, the number of unbroken chain-vertices of type or , the number of even straight chain-vertices, and the number of odd straight chain-vertices. The generating functions for unbroken chains of type L (resp. R) is clearly , the one for even straight chains is , the one for odd straight chains is , and the one for broken chains is . Therefore

Denoting by the total number of chain-vertices and by the total number of straight chain-vertices, this simplifies as

(5)

It is well-known that the generating function of rooted melonic graphs is given by

In addition, a rooted melon-free MO-graph with non-root vertices has edges (recall that the root-edge is split into two edges) where one can insert a rooted melonic graph. Therefore, defining , the generating function of rooted MO-graphs of reduced scheme is given by

(6)

and the generating function of rooted MO-graphs of degree is simply given by

(7)

As discussed in [9], has its main singularity at , , and . Therefore, the dominant terms are those for which is maximized. As shown in Section 6, these schemes are naturally associated to rooted binary trees with inner nodes, as stated in Proposition 6.1. For a fixed rooted binary tree with inner nodes, the total contribution of schemes arising from to the generating function of rooted melon-free MO-graphs of degree is

Indeed, such schemes have , , , each of the non-root leaves is occupied either by the cw or ccw infinity graph, and each of the inner nodes is either occupied by the cycle-graph or the quadruple-edge graph that has possible configurations.

Hence the total contribution to the generating function of rooted MO-graphs is