Counting colored planar maps: algebraicity results
We address the enumeration of properly -colored planar maps, or more precisely, the enumeration of rooted planar maps weighted by their chromatic polynomial and counted by the number of vertices and faces. We prove that the associated generating function is algebraic when is of the form , for integers and . This includes the two integer values and . We extend this to planar maps weighted by their Potts polynomial , which counts all -colorings (proper or not) by the number of monochromatic edges. We then prove similar results for planar triangulations, thus generalizing some results of Tutte which dealt with their proper -colorings. In statistical physics terms, the problem we study consists in solving the Potts model on random planar lattices. From a technical viewpoint, this means solving non-linear equations with two “catalytic” variables. To our knowledge, this is the first time such equations are being solved since Tutte’s remarkable solution of properly -colored triangulations.
Key words and phrases:Enumeration – Colored planar maps – Tutte polynomial – Algebraic generating functions
2000 Mathematics Subject Classification:05A15, 05C30, 05C31
In 1973, Tutte began his enumerative study of colored triangulations by publishing the following functional equation [52, Eq. (13)]:
where stands for (in other words, is the coefficient of in ). This equation defines a unique formal power series in , denoted , which has polynomial coefficients in , and . Tutte’s interest in this series relied on the fact that it “contains” the generating function of properly -colored triangulations of the sphere (Figure 1). More precisely, the coefficient of in is
where the sum runs over all rooted triangulations of the sphere, is the chromatic polynomial of , and the number of faces of .
In the ten years that followed, Tutte devoted at least nine papers to the study of this equation [52, 50, 49, 51, 53, 54, 55, 56, 57]. His work culminated in 1982, when he proved that the series counting -colored triangulations satisfies a non-linear differential equation [56, 57]. More precisely, with and ,
This tour de force has remained isolated since then, and it is our objective to reach a better understanding of Tutte’s rather formidable approach, and to apply it to other problems in the enumeration of colored planar maps. We recall here that a planar map is a connected planar graph properly embedded in the sphere. More definitions will be given later.
We focus in this paper on two main families of maps: general planar maps, and planar triangulations. We generalize Tutte’s problem by counting all colorings of these maps (proper and non-proper), assigning a weight to every monochromatic edge. Thus a typical series we consider is
where runs over a given set of planar maps (general maps or triangulations, for instance), , , respectively denote the number of edges, vertices and faces of , and
counts all colorings of the vertices of in colors, weighted by the number of monochromatic edges. As explained in Section 3, is actually a polynomial in and called, in statistical physics, the partition function of the Potts model on . Up to a change of variables, it coincides with the so-called Tutte polynomial of . Note that is the chromatic polynomial of .
In this paper, we climb Tutte’s scaffolding halfway. Indeed, one key step in his solution of (1) is to prove that, when is of the form
for integers and , then is an algebraic series, that is, satisfies a polynomial equation111Strictly speaking, Tutte only proved this for certain values of and . Odlyzko and Richmond  proved later that his work implies algebraicity for . We prove here that it holds for all and , except those that yield the extreme values . For the polynomials vanish, but we actually weight our maps by , which gives sense to the restriction .
for some polynomial that depends on . Numbers of the form (4) generalize Beraha’s numbers (obtained for ), which occur frequently in connection with chromatic properties of planar graphs [5, 25, 32, 33, 37, 43]. Our main result is that the series defined by (3) is also algebraic for these values of , whether the sum runs over all planar maps (Theorem 15), non-separable planar maps (Corollary 29), or planar triangulations (Theorem 18). These series are not algebraic for a generic value of . In a forthcoming paper, we will establish the counterpart of (2), in the form of (a system of) differential equations for these series, valid for all .
Hence this paper generalizes in two directions the series of papers devoted by Tutte to (1), which he then revisited in his 1995 survey : firstly, because we include non-proper colorings, and secondly, because we study two classes of planar maps (general/triangulations), the second being more complicated than the first. We provide in Sections 12 and 13 explicit results (and a conjecture) for families of 2- and 3-colored maps. Some of them have an attractive form, and should stimulate the research of alternative proofs based on trees, in the spirit of what has been done in the past 15 years for uncolored maps (see for instance [45, 16, 15, 17, 19, 29, 18, 7]). Finally, our results constitute a springboard for the general solution (for a generic value of ), in preparation.
The functional equations we start from are established in Section 4 (Propositions 1 and 2). As (1), they involve two catalytic variables and . Much progress has been made in the past few years on the solution of linear equations of this type [10, 11, 38, 13], but those that govern the enumeration of colored maps are non-linear. In fact, Equation (1) is so far, to our knowledge, the only instance of such an equation that has ever been solved. Our main two algebraicity results are stated in Theorems 15 and 18. In Section 2 below, we describe on a simple example (2-colored planar maps) the steps that yield from an equation to an algebraicity theorem. It is then easier to give a more detailed outline of the paper (Section 2.5). Roughly speaking, the general idea is to construct, for values of of the form (4), an equation with only one catalytic variable satisfied by a relevant specialization of the main series (like in the problem studied by Tutte). For instance, we derive in Section 2 the simple equation (6) from the more complicated one (5). One then applies a general algebraicity theorem (Section 9), according to which solutions of such equations are always algebraic.
Most calculations were done using Maple: several Maple sessions accompanying this paper are available on the second author’s web page (next to this paper in the publication list).
To conclude this introduction, let us mention that the problems we study here have also attracted attention in theoretical physics, and more precisely in the study of models for 2-dimensional quantum gravity. In particular, our results on triangulations share at least a common flavour with a paper by Bonnet and Eynard . Let us briefly describe their approach. The solution of the Potts model on triangulations can be expressed fairly easily in terms of a matrix integral. Starting from this formulation, Daul and then Zinn-Justin [20, 61] used a saddle point approach to obtain certain critical exponents. Bonnet and Eynard went further using the equation of motion method . First, they derived from the integral formulation a (pair of) polynomial equations with two catalytic variables (the so-called loop-equations)222These equations differ from the functional equation (24) we establish for the same problem. But they are of a similar nature, and we actually believe that our method applies to them as well.. From there, they postulated the existence of a change of variables which transforms the loop-equations into an equation occurring in another classical model, the model. The results of [22, 23, 24] on the model then translate into results on the Potts model. In particular, when the parameter of the Potts model is of the form , Bonnet and Eynard obtain an equation with one catalytic variable [21, Eq. 5.4] which may correspond to our equation (42).
2. A glimpse at our approach: properly 2-colored planar maps
The aim of this paper is to prove that, for certain values of (the number of colors), the generating function of -colored planar maps, and of -colored triangulations, is algebraic. Our starting point will be the functional equations of Propositions 1 and 2. In order to illustrate our approach, we treat here the case of properly -colored planar maps counted by edges. It will follow from Proposition 1 that this means solving the following equation:
counts planar maps , weighted by their chromatic polynomial at , by the number of edges and by the degrees and of the root-vertex and root-face (the precise definitions of these statistics are not important for the moment). We are especially interested in the specialization
However, there is no obvious way to derive from (5) an equation for , or even for or . Still, (5) allows us to determine, by induction on , the coefficient of in . The variables and are said to be catalytic.
We can see some readers frowning: there is a much simpler way to approach this enumeration problem! Indeed, a planar map has a proper 2-coloring if and only if it is bipartite, and every bipartite map admits exactly two proper 2-colorings. Thus is simply the generating function of bipartite planar maps, counted by edges. But one has known for decades how to find this series: a recursive description of bipartite maps based on the deletion of the root-edge easily gives:
where . This equation has only one catalytic variable, namely , and can be solved using the quadratic method [30, Section 2.9]. In particular, is found to be algebraic:
What our method precisely does is to reduce the number of catalytic variables from two to one: once this is done, a general algebraicity theorem (Section 9), which states that all series satisfying a (proper) equation with one catalytic variable are algebraic, allows us to conclude. In the above example, our approach derives the simple equation (6) from the more difficult equation (5). We now detail the steps of this derivation.
2.1. The kernel of the equation, and its roots
The functional equation (5) is linear in (though not globally in , because of quadratic terms like ). It reads
where the kernel is
and the right-hand side is:
Following the principles of the kernel method [1, 2, 14, 42], we are interested in the existence of series that cancel the kernel. We seek solutions in the space of formal power series in with coefficients in (the field of fractions in ). The equation can be rewritten
This shows that there exists a unique power series solution (the coefficient of in can be determined by induction on , once the expansion of is known at order ). However, the term having denominator suggests that we will find more solutions if we set , with an indeterminate, and look for in the space of formal power series in with coefficients in . Indeed, the equation now reads (with ):
which shows that there exist two series and that cancel the kernel for this choice of . One of them has constant term 1, the other has constant term . Again, the coefficient of can be determined inductively. Here are the first few terms of and :
Replacing by in the functional equation (7) gives . We thus have four equations,
that relate , , , , and .
We now eliminate from the system (8) the series and the indeterminate to obtain two equations relating , , and . This elimination is performed in Section 6 for a general value of . So let us just give the pair of equations we obtain. The first one is:
Following Tutte , we say that , which takes the same value at and , is an invariant.
Let us denote . The second equation obtained by eliminating and from the system (8) then reads:
where . Then an elementary calculation shows that the identity (10), combined with , implies
We have thus obtained a second invariant 333There is no real need to include the term (which is itself an invariant) in . However, we will see later than this makes a Chebyshev polynomial, a convenient property..
2.3. The theorem of invariants
Consider the invariants (9) and (11) that we have constructed. Both are series in with coefficients in , the field of rational functions in . In , these coefficients are not singular at , except for the coefficient of , which has a simple pole at . We say that has valuation in . Similarly, has valuation in (because of the term ).
Observe that all polynomials in and with coefficients in (the ring of Laurent series in ) are invariants. We prove in Section 8 a theorem — the Theorem of invariants — that says that there are “few” invariants, and that, in particular, must be a polynomial in with coefficients in . Considering the valuations of and in shows that this polynomial has degree 4. That is, there exist Laurent series in , with coefficients in , such that
2.4. An equation with one catalytic variable
In (12), replace by its expression (9) in terms of . The resulting equation involves , , , and five unknown series . The variable has disappeared. Let us now write , and expand the equation in the neighborhood of . This gives the values of the series :
Let us replace in (12) each series by its expression: we obtain
which is exactly the equation with one catalytic variable (6) obtained by deleting recursively the root-edge in bipartite planar maps. It can now be solved using the quadratic method [30, Section 2.9] or its extension (which works for equations of a higher degree in ) described in  and generalized further in Section 9.
2.5. Detailed outline of the paper
With this example at hand, it is easier to describe the structure of the paper. We begin with recalling in Section 3 standard definitions on maps, power series, and the Tutte (or Potts) polynomial. In Section 4 we establish functional equations for -colored planar maps and for -colored triangulations. We then construct a pair ) of invariants in Sections 6 (for planar maps) and 7 (for triangulations). The construction of the invariant is non-trivial, and relies on an independent result which is the topic of Section 5. It is at this stage that the condition naturally occurs. We then prove two “theorems of invariants”, one for planar maps and one for triangulations (Section 8). Applying them provides counterparts of (12), where only one catalytic variable is now involved. Unfortunately, the general algebraicity theorem of  does not apply directly to these equations: we thus extend it slightly (Section 9). In Sections 10 and 11, we prove that this extended theorem indeed applies to the equations with one catalytic variable derived from the theorems of invariants; we thus obtain the main algebraicity results of the paper. Explicit results are given for two and three colors in Sections 12 and 13. Finally, we explain in Section 14 that the algebraicity results obtained for general planar maps imply similar results for non-separable planar maps.
3. Definitions and notation
3.1. Planar maps
A planar map is a proper embedding of a connected planar graph in the oriented sphere, considered up to orientation preserving homeomorphism. Loops and multiple edges are allowed. The faces of a map are the connected components of its complement. The numbers of vertices, edges and faces of a planar map , denoted by , and , are related by Euler’s relation . The degree of a vertex or face is the number of edges incident to it, counted with multiplicity. A corner is a sector delimited by two consecutive edges around a vertex; hence a vertex or face of degree defines corners. Alternatively, a corner can be described as an incidence between a vertex and a face. The dual of a map , denoted , is the map obtained by placing a vertex of in each face of and an edge of across each edge of ; see Figure 2. A triangulation is a map in which every face has degree 3. Duality transforms triangulations into cubic maps, that is, maps in which every vertex has degree 3.
For counting purposes it is convenient to consider rooted maps. A map is rooted by choosing a corner, called the root-corner. The vertex and face that are incident at this corner are respectively the root-vertex and the root-face. In figures, we indicate the rooting by an arrow pointing to the root-corner, and take the root-face as the infinite face (Figure 2). This explains why we often call the root-face the outer face and its degree the outer degree. This way of rooting maps is equivalent to the more standard way, where an edge, called the root-edge, is distinguished and oriented. For instance, one can choose the edge that follows the root-corner in counterclockwise order around the root-vertex, and orient it away from this vertex. The reason why we prefer our convention is that it gives a natural way to root the dual of a rooted map in such a way the root-vertex of becomes the root-face of , and vice-versa: it suffices to draw the vertex of corresponding to the root-face of at the starting point of the arrow that points to the root-corner of , and to reverse this arrow, to obtain a canonical rooting of (Figure 2). In this way, taking the dual of a map exchanges the degree of the root-vertex and the degree of the root-face, which will be convenient for our study.
From now on, every map is planar and rooted. By convention, we include among rooted planar maps the atomic map having one vertex and no edge.
3.2. Power series
Let be a commutative ring and an indeterminate. We denote by (resp. ) the ring of polynomials (resp. formal power series) in with coefficients in . If is a field, then denotes the field of rational functions in , and the field of Laurent series in . These notations are generalized to polynomials, fractions and series in several indeterminates. We denote by bars the reciprocals of variables: that is, , so that is the ring of Laurent polynomials in with coefficients in . The coefficient of in a Laurent series is denoted by , and the constant term by . The valuation of a Laurent series is the smallest such that occurs in with a non-zero coefficient. If , then the valuation is . More generally, for a series , and , we say that has valuation at least in if no coefficient has a pole of order larger than at .
Recall that a power series , where is a field, is algebraic (over ) if it satisfies a non-trivial polynomial equation .
3.3. The Potts model and the Tutte polynomial
Let be a graph with vertex set and edge set . Let be an indeterminate, and take . A coloring of the vertices of in colors is a map . An edge of is monochromatic if its endpoints share the same color. Every loop is thus monochromatic. The number of monochromatic edges is denoted by . The partition function of the Potts model on counts colorings by the number of monochromatic edges:
The Potts model is a classical magnetism model in statistical physics, which includes (when ) the famous Ising model (with no magnetic field) . Of course, is the chromatic polynomial of .
If and are disjoint graphs and , then clearly
If is obtained by attaching and at one vertex, then
The Potts partition function can be computed by induction on the number of edges. If has no edge, then . Otherwise, let be an edge of . Denote by the graph obtained by deleting , and by the graph obtained by contracting (if is a loop, then it is simply deleted). Then
Indeed, it is not hard to see that counts colorings for which is monochromatic, while counts those for which is bichromatic. One important consequence of this induction is that is always a polynomial in and . From now on, we call it the Potts polynomial of . We will often consider as an indeterminate, or evaluate at real values . We also observe that is a multiple of : this explains why we will weight maps by .
Up to a change of variables, the Potts polynomial is equivalent to another, maybe better known, invariant of graphs: the Tutte polynomial (see e.g. ):
where the sum is over all spanning subgraphs of (equivalently, over all subsets of edges) and , and denote respectively the number of vertices, edges and connected components. For instance, the Tutte polynomial of a graph with no edge is 1. The equivalence with the Potts polynomial was established by Fortuin and Kasteleyn :
for . In this paper, we work with rather than because we wish to assign real values to (this is more natural than assigning real values to ). However, we will use one property that looks more natural in terms of : if and are dual connected planar graphs (that is, if and can be embedded as dual planar maps) then
Translating this identity in terms of Potts polynomials thanks to (16) gives:
where and the last equality uses Euler’s relation: .
4. Functional equations
We now establish functional equations for the generating functions of two families of colored planar maps: general planar maps, and triangulations. We begin with general planar maps, for which Tutte already did most of the work. However, he did not attempt, or did not succeed, to solve the equation he had established.
4.1. A functional equation for colored planar maps
Let be the set of rooted maps. For a rooted map , denote by and the degrees of the root-vertex and root-face. We define the Potts generating function of planar maps by:
Since there is a finite number of maps with a given number of edges, and is a multiple of , the generating function is a power series in with coefficients in .
The Potts generating function of planar maps satisfies:
Observe that (1) characterizes entirely as a series in (think of extracting recursively the coefficient of in this equation). Note also that if , then , so that we are essentially counting planar maps by edges, vertices and faces, and by the root-degrees and . The variable is no longer catalytic: it can be set to 1 in the functional equation, which becomes an equation for with only one catalytic variable .
In , Tutte considered the closely related generating function
which counts maps weighted by their Tutte polynomial. He established the following functional equation:
Now, the relation (16) between the Tutte and Potts polynomials and Euler’s relation () give
from which (1) easily follows.
4.2. A functional equation for colored triangulations
Tutte obtained (21) via a recursive description of planar maps involving deletion and contraction of the root-edge. We would like to proceed similarly for triangulations, but the deletion/contraction of the root-edge may change the degrees of the faces that are adjacent to the root-edge, so that the resulting maps may not be triangulations. This has led us to consider a larger class of maps.
We call quasi-triangulations rooted planar maps such that every internal face is either a digon (degree 2) incident to the root-vertex, or a triangle (degree 3). The set of quasi-triangulations is denoted by . It includes the set of near-triangulations, which we define as the maps in which all internal faces have degree 3. For in , we denote by and respectively the number of internal digons and the number of internal digons that are doubly-incident to the root-vertex. For instance, the map of Figure 3(a) satisfies and . A map in is incidence-marked by choosing for each internal digon one of its incidences with the root-vertex. An incidence-marked map is shown in Figure 3(b).
We define the Potts generating function of quasi-triangulations by
As before, denotes the degree of the root-face of . Observe that a map in gives rise to distinct incidence-marked maps. Hence the above series can be rewritten as
where is the set of incidence-marked maps obtained from , and for , the underlying (unmarked) map is denoted .
The Potts generating function of quasi-triangulations, defined by (23), satisfies
where and .
As in the case of general maps, Eq. (24) characterizes the series entirely as a series in (think of extracting recursively the coefficient of in this equation). Moreover, the variable is no longer catalytic when , and the equation becomes much easier to solve. Finally, Tutte’s original equation (1) can be derived from (24), as we explain in Section 14.2.
We first observe that it suffices to establish the equation when , that is, when we do not keep track of the number of edges. Indeed, this number is , by Euler’s relation, so that . Let us thus set .
Equation (15) gives
where the term 1 is the contribution of the atomic map having one vertex and no edge,
where and denote respectively the maps obtained from by deleting and contracting the root-edge .
A. The series . We consider the partition , where (resp. ) is the subset of maps in such that the root-edge is (resp. is not) an isthmus. We denote respectively by and the contributions of and to the generating function , so that
A.1. Contribution of . Deleting the root-edge of a map in leaves two maps in , as illustrated in Figure 4. Hence there is a simple bijection between and the set of ordered pairs of rooted maps in , such that has no internal digon. The Potts polynomial of this pair can be determined using (13). One thus obtains
as is the generating function of maps with no internal digon.
A.2. Contribution of . Deleting the root-edge of a map in gives a map in . Conversely, given , there are at most two ways to reconstruct a map of by adding a new edge and creating a new internal face:
If , one can create an internal triangle,
If , one can create an internal digon; depending on whether the root-edge of is a loop, or not, this new digon will be doubly incident to the root, or not.
In terms of incidence-marked maps, one can create an internal triangle (provided ), or an internal digon marked at its first incidence with the root (provided ), or an internal digon marked at its second incidence with the root (provided the root-edge of is a loop). These three possibilities are illustrated in Figure 5. In the third case, the map is obtained by gluing at the root two maps and such that has outer degree 1, and is easily determined using (14). This gives:
as is the generating function of maps with outer degree .
B. The series . We now consider the partition , where (resp. ) is the subset of maps in such that the root-edge is (resp. is not) a loop. We denote respectively by and the contributions of and to the generating function , so that
B.1. Contribution of . Contracting the root-edge of a map in is equivalent to deleting this edge. It gives a map in , formed of two maps of attached at a vertex. Hence there is a simple bijection, illustrated in Figure 6, between and the set of ordered pairs of rooted maps in , such that the map has outer degree 1 or 2. The Potts polynomial of the map obtained by gluing and can be determined using (14). One thus obtains
The factor 2 accounts for the two ways of marking incidences in the new digon that is created when has outer degree 1.
B.2. Contribution of Contracting the root-edge of a map in gives a map that may not belong to , as contraction may create faces of degree 1. This happens when the face to the left of is an internal digon (Figure 7).
For a map in , we consider the maximal sequence of edges , such that is the root-edge and for , the edges and bound an internal digon. We partition further, writing , depending on whether the face to the left of is the outer face, or not. We consistently denote by and the respective contributions of these sets to .
B.2.1. Contribution of . As shown on the left of Figure 7, there is a bijection between the set and the set of triples , where and , are maps in such that has no internal digon. Let