Enumeration of Planar Rooted Triangulations
Jason Zhicheng Gao
School of Mathematics and Statistics, Carleton University
June 8, 2000
[summary by Gilles Schaeffer]
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This talk presents a joint work with I. M. Wanless and
N. C. Wormald [5
A planar map is a connected graph embedded in the plane. In
this talk, loops and multiple edges are forbidden. A map is
rooted if one edge is oriented. The start point of this root
is called the root vertex, the face on its right the root
face and the face on its left the near face. If the root
and near face of a planar map are the same, the root is a bridge. By
convention the root is always taken so that the root face is the
infinite face. The other faces are then called interior faces.
The degree of a face is its number of incidences of edges
(i.e., bridges count for two). A triangulation is a map whose
faces all have degree three. A map is p-connected if at
least p vertices must be removed to separate it into two connected
Euler already enumerated triangulations of polygons in the 19th
century (they are Catalan), but the enumeration of triangulations
as defined here started with Tutte's work in the sixties. Several
families of planar rooted triangulations were in fact enumerated:
All these family of planar rooted triangulations have algebraic
generating functions and asymptotic behaviors of the same form,
4-connected triangulations (see :
algebraic generating function and asymptotic are given),
- 3-connected triangulations (see [2, 8]:
with n vertices they are (4n+1)!/(n+1)! (3n+2)!),
- 2-connected allowing multiple edges (see :
with n vertices they are 3·2n(3n)!/n! (2n+2)!),
- all triangulations allowing loops and multiple edges
(see : algebraic generating function and asymptotic are
where n denotes the number of vertices. For connectivity i from 1
to 4, the values of ri are
(3)1/2/36, 2/27, 27/256, 4/27,
respectively. For no planar map is 6-connected, the only missing
connectivity for triangulations was 5, which is the subject of the
present study: it turns out that for 5-connected triangulations, the
generating function is algebraic of degree 6, and the asymptotic
behavior is similar, with r5 given as a root of a certain
polynomial P of degree 6 such that
It is amusing to remark that r5 is not the smallest positive
root the polynomial P.
The proof is based on skillful refinements of the three original
ingredients of Tutte's method: root edge deletion, the quadratic
method, and composition schemes.
2 Root Edge Deletion
The deletion of the root edge is maybe the simplest possible idea to
decompose a map. It turns out to be very efficient in providing
functional equation for generating functions of ``not-too-connected''
In general there are two cases in the root edge deletion process
applied to a planar map M of a family F:
This decomposition can be made one-to-one, at the expense of taking
the root face degree into account. It then results into functional
equations for the generating function
where fn,k denote the number of maps with n inner vertices and
a root face of degree k.
either the root edge deletion separates M into two pieces that
more or less belong to F,
- or it yields directly a map M' that belongs more or less to
the family F. In this case, M' usually has a larger root face
degree than M: the removal of the root has merged the root and near
faces of M.
For instance let F(x,y) be the generating function of
near-triangulations, i.e., maps with all faces of degree three,
except maybe the root face. Then the root edge deletion yields
where F3(x) is the generating function of triangulations, i.e.,
Indeed in the right hand side, the three summands correspond to three
cases in the decomposition of a near-triangulation M:
In order to enumerate 5-connected triangulations, it turns out to be
necessary to enumerate M-type maps, i.e., maps whose interior
faces have degree three or four. Their generating function
where mn,l,k denotes the number of rooted M-type maps with
n triangular interior faces, l interior quadrangular faces and a
root face of degree k, satisfies
M is the degenerate triangulation with one edge and two vertices,
- M is made of a couple of triangulations separated by a rooted
- or removing the root of M directly yields a triangulation
M'. In this case, M' must not be the degenerate triangulation, nor
have a short diagonal cutting it into a triangulation and a
near-triangulation (otherwise, replacing the root of M would create
a double edge).
where M3 and M4 denote the generating functions of M-type maps
with root face of degree three and four respectively.
The root edge deletion applied to M-type maps yields, with M'=M-1,
3 The Quadratic Method
The equations provided by root edge deletion have always the same
flavor: they involve a principal generating function (F(y) for
near-triangulations) in which the equation is quadratic, and a
secondary generating function not depending on y (F3 for
The quadratic method, as used by Tutte, proceeds as follows
In the case of M-type maps, the situation is somewhat more involved,
because of the presence of two secondary generating functions. However
using a theorem of Brown on power series that are square
roots , Bender and Canfield have dealt with a similar
situation in . Upon finding appropriate parametrizations
to make the computation tractable with Maple, this approach
Write the equation in the form A2=B, where A and B are
polynomials in all variables and generating functions and where B does not
contain the principal generating function. E.g., for near-triangulations this
In general this is possible because the equation is quadratic in F.
||(x-xyF3-y), and B=
- Show that there exists a power series Y(x) such that
and, provided this system is not degenerate, this proves that F3
and Y are algebraic.
M3=u3-2uv+u, and M4=3u4-5u2v+u2-v2+v+2,
where u=u(x,y) and v=v(x,y) are the power series uniquely
4 Composition Schemes and Non-Uniqueness
Root edge deletion does not work well on 4-connected triangulations or
triangulations with higher connectivity, because the deletion of the
root can produce maps with smaller connectivity that are hard to
decompose back into maps with high connectivity. To enumerate
4-connected triangulations, Tutte introduced compositions schemes.
First remark that a triangulation is 3-connected as soon as it
contains no loop and multiple edges, 4-connected if all its cycles of
length three bound faces, and 5-connected if moreover it contains no
4-cycles with a vertex inside.
In the last two sections we were able to determine the generating
function F3(x) of (3-connected) triangulations. Now take a
3-connected triangulation M. Its cycles of length three are ordered
by inclusion. In particular they are all inside the outer cycle of
the root face; call a cycle of length three maximal if it is
not inside any other one. A maximal cycle either bounds a face or
contains at least one vertex in its inside. In the latter case, the
maximal cycle and its inside form a triangulation.
Removing the triangulation inside each maximal cycle yields the
decomposition of M into a 4-connected triangulation M' plus
one triangulation per face of M' (possibly reduced to a triangle).
In terms of generating functions, this yields
where Gk is the number of 4-connected triangulations with
k vertices (and 2k+1 inner faces). This yields a fonctional
equation of the composition type
which properly determines the generating function G(a) in terms of
f(x)=F3(x)2. Indeed, consider the equation a=xf(x). As
f(x)=1+O(x) this equation properly defines a power series x(a) and
from the composition equation,
Now as F3(x) is algebraic, so is f(x) and there is a
polynomial P(x,f) such that P(x,f(x))=0. Take x=x(a) so that
and we conclude that x(a), and thus G(a), are algebraic.
The next step is to go from 4-connected triangulation to 5-connected
ones. The idea is again to start with a triangulation and remove the
inside of any non empty cycle of length three or four. However in
general this yields an M-type map and not a triangulation.
The composition scheme has thus to be defined between 4- and
5-connected M-type maps. The same technique immediately applies to
remove cycles of length three in M-type maps, but for cycle of length
four, a new difficulty appear: two four cycles can overlap, making the
definition of maximal four-cycles no so easy.
Finally, it turns out that a careful case study allows to classify
overlapping four-cycles and work out the desired composition schemes.
Using the latter composition scheme and the results for M3(x,y)
and M4(x,y), algebraic equations for the generating function T(x)
of 5-connected triangulations can finally be derived. These equations
take the form of a parametrization T(x)=F(x,s) where s has a
relatively compact algebraic equation (unlike T(x)).
The asymptotic is then obtained from a careful analysis of the
possible sources of singularity in the parametrization. This indirect
approach seems more easily tractable than dealing with the explicit
polynomial equation giving T(x).
This concludes the story for planar triangulations. As far as exact
expression for generating functions are concerned, for general planar
maps, there is no more than Tutte's result giving 3-connected
ones. On higher genus surfaces, 2-connectivity was the limit until the
very recent result of  for 3-connected triangulations of
the projective plane.
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vol. 170, 1967, pp. 327--333.
Gao (Z. C.) and Wang (J. Y.). --
Exact enumeration of rooted 3-connected triangular maps on the
projective plane. --
Gao (Z. C.), Wanless (I. M.), and Wormald (N. C.). --
Counting 5-connected planar triangulations. --
Gao (Zhi-Cheng). --
The number of rooted triangular maps on a surface. Journal of
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Mullin (R. C.). --
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Tutte (W. T.). --
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