uniﬁed construction between blossoming trees and...

Unified construction betweenblossoming trees and planar maps

Shanghai Jiao Tong University, November 23rd 2013

Marie Albenque (CNRS, Paris)Joint work with Dominique Poulalhon

A plane map is the embedding of a connected graph in the planeup to continuous deformations.

Plane Maps.


Plane Maps.

plane map = plane graph + cyclic order of neigbours around each vertex.


Plane Maps.

Faces = connected components of the plane when the edge are removed



Plane Maps.

Plane maps are rooted.



Plane maps are rooted.


Plane Maps.


There is one special face which is infinite: the outer face.


Some families of maps

Simple triangulationsfaces are trianglesno loops nor multiple edges



4-regular mapsall vertices have degree 4



4-regular mapsall vertices have degree 4

Bipolar orientationsacyclic orientationonly one sink tonly one source s

s

t

Why maps ?

What the motivation for studying maps instead of graphs ?

Because maps have more structure than graphs,they are actually simpler to study.

Why maps ?



Euler Formula : # vertices+#faces = 2+#edges

A triangulation with 2n faces has 3n edges and n+ 2 vertices.

Why maps ?



Structure allows recursive decomposition ) enumeration [Tutte, ’60s].

Euler Formula : # vertices+#faces = 2+#edges

A triangulation with 2n faces has 3n edges and n+ 2 vertices.


Simple triangulations

4-regular maps

Bipolar orientations



4-regular maps


Number of simple triangulations with n+ 2 vertices:

�n =2(4n� 3)!n!(3n� 1)!

[Tutte ’62], [Poulalhon, Schae↵er 05]



4-regular maps


Number of rooted 4-regular maps with n vertices:


�n =2(4n� 3)!n!(3n� 1)!


[Tutte ’62], [Schae↵er ’97]

Rn =2 · 3n

(n+ 1)

2nn

!



4-regular maps

Bipolar orientationsNumber of bipolar orientations with i+ 2 vertices and j + 1 faces:

[Baxter ’01], [Fusy, Poulalhon, Schae↵er’09],

[Bonichon, Bousquet-Melou, Fusy ’10]

⇥ij =2(i+ j)!(i+ j + 1)!(i+ j + 2)!i!(i+ 1)!(i+ 2)!j!(j + 1)!(j + 2)!

.

Number of rooted 4-regular maps with n vertices:


�n =2(4n� 3)!n!(3n� 1)!


[Tutte ’62], [Schae↵er ’97]

Rn =2 · 3n

(n+ 1)

2nn

!

Enumeration

One of the main question when studying some families of maps:

How many maps belong to this family ?

• Recursive decomposition: Tutte ’60s. Baxter ’01

• Matrix integrals: t’Hooft ’74,Brezin, Itzykson, Parisi and Zuber ’78.

• Representation of the symmetric group: Goulden and Jackson ’87.

• Bijective approach with labeled trees: Cori-Vauquelin ’81, Schae↵er’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy,Miermont, ...

• Bijective approach with blossoming trees: Schae↵er ’98, Schae↵erand Bousquet-Melou ’00, Poulalhon and Schae↵er ’05, Fusy, Poulalhonand Schae↵er ’06.

Bijective proofs ?



Bijective proofs ?



Bijective proof give enumeration formulas but they also yield:

• an explanation of why the formulas are so simple.

• fast algorithms to encode/generate random maps

• better understanding of the structure of the map.

What is a blossoming tree ?

A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems


A blossoming tree is a plane tree where vertices can carryopening stems or closing stems, such that :

# closing stems = # opening stems

A plane map can be canonically associated to any blossoming tree bymaking all closures clockwise.


If the tree is rooted and its edges oriented towards the root + closureedges oriented naturally

) Accessible orientation of the map without ccw cycles.

A plane map can be canonically associated to any blossoming tree bymaking all closures clockwise.

Theorem : [A., Poulalhon]If a plane map M with a marked vertex v is endowed with anorientation such that :• there exists a directed path from any vertex to v,• there is no counterclockwise cycle,

then there exists a unique blossoming tree rooted at v whose closureis M endowed with the same orientation.

Can we always find a blossoming tree from a plane map ?

Theorem : [A., Poulalhon]If a plane map M with a marked vertex v is endowed with anorientation such that :• there exists a directed path from any vertex to v,• there is no counterclockwise cycle,

then there exists a unique blossoming tree rooted at v whose closureis M endowed with the same orientation.

Can we always find a blossoming tree from a plane map ?

Proof by induction onthe number of faces +identification of closureedges ....

Orientations

To apply the construction: need to find canonical orientations

Orientation = orientation of the edges of the map

Orientations



canonical orientation =

acyclic + accessible


Orientations





4-regular maps

2 outgoing edges/vertex

2 ingoing edges/vertex


Orientations





4-regular maps




A map is 4-regular i↵ it admits an orientation withindegree 2 and outdegree 2 for each vertex.

Orientations





4-regular maps




3 outgoing edges/ non-root vertex

1 outgoing edge / root vertex


Orientations





4-regular maps







A triangulation is simple i↵ it admits an orientation with:outdegree 3 for each non-root vertex,outdegree 1 for each vertex on the root face.

Orientations

Many families of maps admit a caracterization via orientations




4-regular maps






Orientations




4-regular maps






Theorem requires accessible orientation without ccw cycles :Too much to ask ?

Orientations




4-regular maps







NO !

Orientations




4-regular maps







NO !

Proposition: [Felsner ’04]For a given map and orientation, there exists a unique orientation withthe same outdegrees and without ccw cycles.If there exists one accessible ↵-orientation, all of them are accessible.

Summary

• Take a family of maps,

Maps with even degrees

Summary



• Try to find a caracterization of the family by an orientation,

Orientations with same out/in degrees

Summary





• Consider the unique orientation without counterclockwise cycles,

Summary



• Consider the unique orientation without counterclockwise cycles,• Try to find a caracterization of the family by an orientation,


Summary



• Consider the unique orientation without counterclockwise cycles,• Apply the bijection,



Summary



• Consider the unique orientation without counterclockwise cycles,• Apply the bijection,• Study the family of blossoming trees.


Orientations with same out/in degreesTrees with same out/indegrees

Blossoming trees and bipolar orientations

s

t


s

t

s

t


s

t t

marked vertex 2 outer face ) easy to compute the blossoming tree

[Bernardi ’07]


s

t t

marked vertex 2 outer face ) easy to compute the blossoming tree

[Bernardi ’07]Description/enumeration of these trees ?

Bijection

Blossoming trees and triplet of paths

blossoming trees obtained after opening a bipolarorientation with i+ 2 vertices and j + 1 faces

Tbip(i, j) =



Tbip(i, j) =

Theorem

=blossoming trees which closes into a bipolar orientationwith i+ 2 vertices and j + 1 faces



Tbip(i, j) =

Theorem

Proposition: [A., Poulalhon]There exists a one-to-one correspondence between :

Tbip(i, j) and

triplet of non-intersecting paths

with i and j

and fixed first and final points

=blossoming trees which closes into a bipolar orientationwith i+ 2 vertices and j + 1 faces



Tbip(i, j) =


Tbip(i, j) and


with i and j


Lindstrom-Gessel-Viennot Lemma

#Tbip(i, j) = ⇥ij =2(i+j)!(i+j+1)!(i+j+2)!

i!(i+1)!(i+2)!j!(j+1)!(j+2)!

Trees of Tbip

Trees of Tbip

• First son of the root = only opening stems

Trees of Tbip


• Closures must not wrap the root

Trees of Tbip



• Opening stem cannot close into its subtree

Trees of Tbip




• Leaf of the tree ( 6= s) must carry one closing stem.

Trees of Tbip





Encoding of the blossoming tree = contour word = word on {e, e, b, b} s.t.:

e, e : first time, second time we see an edgeb, b : opening stem, closing stem.

w = e b b b e e e e b e e b b b e e b b e e b b e b e b b e

Trees of Tbip








) w = eb

Trees of Tbip








) w = eb

) w[b, b] = Dyck word

Trees of Tbip








) w = eb

) w[b, b] = Dyck word

) w = e . . . e . . . b . . . e . . .

) w = . . . b . . . e . . . b . . . 9

9

Trees of Tbip and triple of paths

w = eb, w[e, e] and w[b, b] = Dyck words,w = . . . b . . . e . . . b . . . , w = e . . . e . . . b . . . e . . .





w1 := w[e, b], w2 := w[e, b]




w1 := w[e, b], w2 := w[e, b]

w1 = e e e e b e b b e b b e b bw2 = e b e b e e b e b e b b b e

e

e

b

b




w1 := w[e, b], w2 := w[e, b]


e

e

b

b • w2 = on the right/bottom of w1

• no vertical edge in common.

• w1 = e . . . b w2 = eb

+




w1 := w[e, b], w2 := w[e, b]


e

e

b



• w1 = e . . . b w2 = eb

+

e

eb

b



w1 := w[e, b], w2 := w[e, b]


e

e

b



• w1 = e . . . b w2 = eb

+

e

eb

b

+ w3 = w[e, b] = triple of paths !

Summary


Tbip(i, j) and


with i and j


Corollary : The number ⇥ij of bipolar orientationswith i+ 2 vertices and j + 1 faces is equal to:

⇥ij =2(i+ j)!(i+ j + 1)!(i+ j + 2)!

i!(i+ 1)!(i+ 2)!j!(j + 1)!(j + 2)!.

Conclusion + perspectives

Our framework can be applied to many families of maps :• Simple triangulations and quadrangulations• Eulerian and general maps• Non-separable maps• Constellations• Plane bipolar orientations• d-angulations of girth d• . . .

New bijectionsNew bijections

simpler proofs ofknown bijections





Unified point of view + can work even if no closed formulas






Di↵erent version of a unified bijection in [Bernardi, Fusy ’10],

• Can we make a connection between both ?

• Can we understand the bijections better: distance, ...





Thank you !


Di↵erent version of a unified bijection in [Bernardi, Fusy ’10],

• Can we make a connection between both ?

• Can we understand the bijections better: distance, ...

uniﬁed construction between blossoming trees and...

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