A master bijection for planar maps and its applications Olivier Bernardi (MIT) Joint work with ´ Eric Fusy (CNRS/LIX) UCLA, March 2011
Planar maps. Definition A planar map is a connected planar graph embedded in the sphere considered up to continuous deformation. = � =
Planar maps. Motivations • Algorithmic applications: efficient encoding of meshed surfaces. Appears courtesy to Wikipedia • Probability and Physics: random lattices, random surfaces. Appears courtesy to G. Chapuy • Representation Theory: factorization problems.
Planar maps. Methods • Generating functions [Tutte 63] Recursive description of maps � recurrences. • Matrix Integrals [’t Hooft 74] Feynmann Diagram ≈ maps. • Representation theory [Jackson 90] Factorizations of permutations ≈ maps. • Bijections [Cori-Vauquelin 91, Schaeffer 98] Maps � decorated trees.
Planar maps. Counting results Planar maps. Counting results • Triangulations ( 2 n faces) 2 n 1 � 4 n − 2 � 3 n � � Simple: Loopless: n (2 n − 1) n − 1 ( n + 1)(2 n + 1) n • Quadrangulations ( n faces) � 3 n 2 · 3 n � 2 n 2 � � General: Simple: ( n + 1)( n + 2) n ( n + 1) n − 1 n • Bipartite maps ( n i faces of degree 2 i ) 2 · ( � i n i )! � n i 1 � 2 i − 1 � (2 + � ( i − 1) n i )! n i ! i i Common features: Algebraic generating function. Asymptotic ∼ κ n − 5 / 2 R n .
Planar maps. Counting results • Triangulations ( 2 n faces) 2 n 1 � 4 n − 2 � 3 n � � Simple: Loopless: n (2 n − 1) n − 1 ( n + 1)(2 n + 1) n [PoSc06] [FuPoSc08] [ScPo02] [B.07] • Quadrangulations ( n faces) � 3 n 2 · 3 n � 2 n 2 � � General: Simple: ( n + 1)( n + 2) n ( n + 1) n − 1 n [Sc98] [Fu07] [Sc97] [Sc98] • Bipartite maps ( n i faces of degree 2 i ) 2 · ( � i n i )! � n i 1 � 2 i − 1 � (2 + � ( i − 1) n i )! n i ! i i [Sc97] [BoDiGu04] Abbreviations: Bo: Bouttier Gu: Guitter Di: Di Francesco Po: Poulalhon Fu: Fusy Sc: Schaeffer
Outline Describe a master bijection for planar maps which generalizes all the known bijections (of the red type). Benefits : • Simplifies/unifies the proofs. • Helps to find bijections for new classes of maps.
Outline 1. Master bijection between a class of oriented maps and a class of bicolored decorated trees . 2. Specializations to classes of maps (via canonical orientations). Girth 4 [Sc98] 3 [FuPoSc08] 2 [PoSc02] 1 [Sc98,BoDiGu04] Degree of the faces 6 7 8 1 2 3 4 5
Master bijection (simplified version)
Plane orientations. A plane orientation is a oriented map drawn in the plane.
Plane orientations. A plane orientation is a oriented map drawn in the plane. We consider the set O of plane orientations which are: • minimal : there is no counterclockwise directed cycle, • accessible : any internal vertex can be reached from an external vertex, and have external vertices of indegree 1.
Mobiles A mobile is a plane tree with vertices properly colored in black and white, together with buds (half-edges) incident to black vertices. The excess is the number of edges minus the number of buds.
Master bijection Mapping Φ for a plane orientation O in O : • Return the external face. • Place a black vertex v f inside each internal face f . Turning clockwise around f , draw an edge/bud from v f to the corners following forward/backward edges • Erase the edges of O and its external vertices.
Master bijection Theorem [B.,Fusy]: The mapping Φ is a bijection between the set O of plane orientations and the set of mobiles with negative excess. Moreover, indegree of internal vertices ← → degree of white vertices degree of internal faces ← → degree of black vertices degree of external face ← → - excess
Using the master bijection to count classes of maps? Strategy for counting a class C of maps: • Define a “canonical” orientation in O for each map in C . This identifies C with a subset O C of O . • Characterize (and count) the set of mobiles associated to O C via the master bijection Φ .
Using the master bijection to count classes of maps? How to define an orientation in the set O (orientations minimal, ac- cessible with external vertices of indegree 1)?
Using the master bijection to count classes of maps? How to define an orientation in the set O (orientations minimal, ac- cessible with external vertices of indegree 1)? Let G = ( V, E ) be a map and let α : V → N . Fact 1 : If there exists an orientation of G with indegrees α ( v ) , then there exists a unique minimal one. ⇒ Orientations O can be defined by specifying the indegrees.
Using the master bijection to count classes of maps? How to define an orientation in the set O (orientations minimal, ac- cessible with external vertices of indegree 1)? Let G = ( V, E ) be a map and let α : V → N . Fact 1 : If there exists an orientation of G with indegrees α ( v ) , then there exists a unique minimal one. ⇒ Orientations O can be defined by specifying the indegrees. Fact 2 : There exists an orientation with indegrees α ( v ) if and only if • | E | = � v ∈ V α ( v ) | E U | ≤ � • for any U ⊂ V , v ∈ U α ( v ) .
Using the master bijection to count classes of maps? How to define an orientation in the set O (orientations minimal, ac- cessible with external vertices of indegree 1)? Let G = ( V, E ) be a map and let α : V → N . Fact 1 : If there exists an orientation of G with indegrees α ( v ) , then there exists a unique minimal one. ⇒ Orientations O can be defined by specifying the indegrees. Fact 2 : There exists an orientation with indegrees α ( v ) if and only if • | E | = � v ∈ V α ( v ) | E U | ≤ � • for any U ⊂ V , v ∈ U α ( v ) . Moreover, the orientation is accessible if and only if | E U | < � v ∈ U α ( v ) whenever U does not contain all the external vertices.
Using the master bijection to count classes of maps? Conclusion : For a map G = ( V, E ) , one can define an orientation in O by specifying a function α : V → N such that: • α ( v ) = 1 for every external vertex v , (*) • | E | = � v ∈ V α ( v ) , • ∀ U ⊂ V , | E U | ≤ � v ∈ U α ( v ) with strict inequality if U does not contain all external vertices.
Using the master bijection to count classes of maps? Conclusion : For a map G = ( V, E ) , one can define an orientation in O by specifying a function α : V → N such that: • α ( v ) = 1 for every external vertex v , (*) • | E | = � v ∈ V α ( v ) , • ∀ U ⊂ V , | E U | ≤ � v ∈ U α ( v ) with strict inequality if U does not contain all external vertices. Remark : Specifying an orientation in O by the indegrees of internal vertices is also convenient in view of applying the master bijection Φ : indegrees of internal vertices ← → degrees of white vertices.
Example of specialization: simple triangulations Girth 4 3 2 1 Degree of faces 6 7 1 2 3 4 5
Triangulations Fact: A triangulation with n internal vertices has 3 n internal edges. Proof: The numbers v , e , f of vertices edges and faces satisfy: • Incidence relation: 3 f = 2 e . • Euler relation: v − e + f = 2 . �
Triangulations Fact: A triangulation with n internal vertices has 3 n internal edges. Natural candidate for indegree function: � 3 if v internal � α : v �→ 1 if v external . � � 1 1 3 3 3 3 1
Triangulations Thm [Schnyder 89]: A triangulation admits an orientation with in- degree function α if and only if it is simple. New (easier) proof: Use the Euler relation + the incidence relation to show that α satisfies condition (*) . �
Triangulations Thm [Schnyder 89]: A triangulation admits an orientation with in- degree function α if and only if it is simple. ⇒ The class T of simple triangulations is indentified with the class of plane orientation O T ⊂ O with faces of degree 3, and internal vertices of indegree 3. Thm [recovering FuPoSc08]: By specializing the master bijection Φ to O T one obtains a bijection between simple triangulations and mobiles such that • black vertices have degree 3 • white vertices have degree 3 • the excess is − 3 (redundant).
Triangulations Counting: The generating function of mobiles with vertices of degree 3 rooted on a white corner is T ( x ) = U ( x ) 3 , where U ( x ) = 1 + xU ( x ) 4 . Consequently, the number of (rooted) simple triangulations with 2 n 1 � 4 n − 2 � faces is . n (2 n − 1) n − 1
More specializations: d -angulations of girth d . Girth 4 3 2 1 Degree of faces 6 7 1 2 3 4 5
d -angulations of girth d Fact: A d -angulation with ( d − 2) n internal vertices has dn internal edges. d = 5
d -angulations of girth d Fact: A d -angulation with ( d − 2) n internal vertices has dn internal edges. Natural candidate for indegree function: � d/ ( d − 2) if v internal � α : v �→ . . . � 1 if v external � d = 5
d -angulations of girth d Fact: A d -angulation with ( d − 2) n internal vertices has dn internal edges. Idea: We can look for an orientation of ( d − 2) G with indegree function � d if v internal � α : v �→ 1 if v external . � � 5 5 5 d = 5 5 5 5
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