Ramified coverings of the sphere by itself (Cont’d) A mapping φ is a ramified covering of S by S if there exists a finite subset X = { x 1 , . . . , x p } such that: • φ S \ φ − 1 ( X ) is a covering, and D = S • φ is ramified over each x i I = S regular value critical value critical value
Ramified coverings of the sphere by itself (Cont’d) A mapping φ is a ramified covering of S by S if there exists a finite subset X = { x 1 , . . . , x p } such that: • φ S \ φ − 1 ( X ) is a covering, and D = S • φ is ramified over each x i On each component V j of φ − 1 ( V ( x i )) , for some integer λ ( i ) φ ∼ φ λ ( i ) j . j I = S regular value critical value critical value λ (2) = 1 , 22 λ (2) = 2 , 3 λ (1) = 15
Ramified coverings of the sphere by itself (Cont’d) A mapping φ is a ramified covering of S by S if there exists a finite subset X = { x 1 , . . . , x p } such that: id id • φ S \ φ − 1 ( X ) is a covering, and id D = S φ 3 • φ is ramified over each x i generically n φ 2 sheets id On each component V j of φ − 1 ( V ( x i )) , id for some integer λ ( i ) φ ∼ φ λ ( i ) j . φ 2 φ 2 j id I = S regular value critical value critical value λ (2) = 1 , 22 λ (2) = 2 , 3 λ (1) = 15
Ramified coverings of the sphere by itself (Cont’d) A mapping φ is a ramified covering of S by S if there exists a finite subset X = { x 1 , . . . , x p } such that: id id • φ S \ φ − 1 ( X ) is a covering, and id D = S φ 3 • φ is ramified over each x i generically n φ 2 sheets id On each component V j of φ − 1 ( V ( x i )) , id for some integer λ ( i ) φ ∼ φ λ ( i ) j . φ 2 φ 2 j id The ramification type over a critical value x i is the partition λ ( i ) I = S regular value critical value critical value The passport of a ramified covering is the list Λ = ( λ (1) , . . . , λ ( p ) ) λ (2) = 1 , 22 λ (2) = 2 , 3 λ (1) = 15 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) id id id φ 3 generically n φ 2 sheets id id φ 2 φ 2 id D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its id id preimages. id φ 3 generically n φ 2 sheets id id φ 2 φ 2 id D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D but if we wind around critical points D = S I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D but if we wind around critical points D = S some sheets may get permuted I = S regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D but if we wind around critical points D = S some sheets may get permuted I = S • visiting critical points create multiple values or ”vertices” regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D but if we wind around critical points D = S some sheets may get permuted I = S • visiting critical points create multiple values or ”vertices” regular value critical value critical value λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering
Ramified coverings of the sphere by itself (Cont’d) To understand the ”shape” of the covering, draw paths on I and study its preimages. • n independant preimages as long as we stay away from critical points • a contractible loop on I yields n contractible loops on D but if we wind around critical points D = S some sheets may get permuted I = S • visiting critical points create multiple values or ”vertices” regular value critical value critical value ⇒ The partitions λ ( i ) λ (1) = 15 λ (2) = 1 , 22 λ (2) = 2 , 3 are partitions of n , the passport Λ = ( λ (1) , . . . , λ ( p )) of a ramified covering degree of the covering.
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 2 1 D = S I = S
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 D = S I = S
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 Contractible loop in S \ X D = S ⇒ identity permutation I = S
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 Contractible loop in S \ X D = S ⇒ identity permutation I = S Concatenation of two loops ⇒ product of the permutations Example: (1)(2 , 3 , 4 , 5) · (1 , 2)(3 , 4)(5)
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 Contractible loop in S \ X D = S ⇒ identity permutation I = S Concatenation of two loops ⇒ product of the permutations Example: (1)(2 , 3 , 4 , 5) · (1 , 2)(3 , 4)(5)
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 Contractible loop in S \ X D = S ⇒ identity permutation I = S Concatenation of two loops ⇒ product of the permutations Example: (1)(2 , 3 , 4 , 5) · (1 , 2)(3 , 4)(5)
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 Contractible loop in S \ X D = S ⇒ identity permutation I = S Concatenation of two loops ⇒ product of the permutations Example: (1)(2 , 3 , 4 , 5) · (1 , 2)(3 , 4)(5) ⇒ Equivalence classes of ramified coverings ≡ factorizations of permutations
Monodromy, and permutations Let us label { 1 , . . . , n } the preimages of a regular point. Loop ⇒ permutation of sheet labels 5 Example: (1 , 2)(3 , 4)(5) in cyclic notation 4 3 The permutation is invariant under continuous deformation of the loop 2 provided it stays in S \ { X } 1 Contractible loop in S \ X D = S ⇒ identity permutation I = S Concatenation of two loops ⇒ product of the permutations Example: (1)(2 , 3 , 4 , 5) · (1 , 2)(3 , 4)(5) ⇒ Equivalence classes of ramified coverings ≡ factorizations of permutations but geometric intuition is lost
coverings with 3 critical values and bipartite maps D = S 2 1 1 1 2 2 1 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values
coverings with 3 critical values and bipartite maps D = S 2 4 2 1 1 5 6 1 7 2 1 3 2 1 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
coverings with 3 critical values and bipartite maps On I , draw an edge between • and ◦ via the basepoint D = S 2 4 2 1 1 5 6 1 7 2 1 3 2 1 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
coverings with 3 critical values and bipartite maps On I , draw an edge between • and ◦ via the basepoint D = S 2 4 We get a planar map: 2 1 1 5 6 that is, a graph embedded on 1 the sphere with simply 7 2 1 3 connected faces 2 1 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
coverings with 3 critical values and bipartite maps On I , draw an edge between • and ◦ via the basepoint D = S 2 4 We get a planar map: 2 1 1 5 6 that is, a graph embedded on 1 the sphere with simply 7 2 1 3 connected faces 2 1 8 1 Proof. Faces are simply connected because a loop around the edge in I can be deformed to a loop around � 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
coverings with 3 critical values and bipartite maps On I , draw an edge between • and ◦ via the basepoint D = S 2 4 We get a planar map: 2 1 1 5 6 that is, a graph embedded on 1 the sphere with simply 7 2 1 3 connected faces 2 1 8 1 Proof. Faces are simply connected because a loop around the edge in I can be deformed to a loop around � 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
coverings with 3 critical values and bipartite maps On I , draw an edge between • and ◦ via the basepoint D = S 2 4 We get a planar map: 2 1 1 5 6 that is, a graph embedded on 1 the sphere with simply 7 2 1 3 connected faces 2 1 8 1 Proof. Faces are simply connected because a loop around the edge in I can be deformed to a loop around � 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
coverings with 3 critical values and bipartite maps On I , draw an edge between • and ◦ via the basepoint D = S 2 4 We get a planar map: 2 1 1 5 6 that is, a graph embedded on 1 the sphere with simply 7 2 1 3 connected faces 2 1 8 1 Proof. Faces are simply connected because a loop around the edge in I can be deformed to a loop around � 2 1 Proposition. This is a bijection between bipartite planar maps I = S and ramified coverings of S by S with 3 critical values. λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S 2 4 2 1 1 5 6 1 7 2 1 3 2 1 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 1 5 6 1 7 2 1 3 2 1 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 σ • = (1)(2 , 6)(3 , 5)(4 , 7)(8) 1 5 6 with cyclic type λ • 1 7 2 1 3 Cycle types ⇔ degree 2 1 distributions 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 σ • = (1)(2 , 6)(3 , 5)(4 , 7)(8) 1 5 6 with cyclic type λ • 1 7 2 1 3 Cycle types ⇔ degree 2 1 distributions 8 1 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 σ • = (1)(2 , 6)(3 , 5)(4 , 7)(8) 1 5 6 with cyclic type λ • 1 7 2 1 3 Cycle types ⇔ degree 2 1 distributions 8 1 What about σ � and λ � ? 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 σ • = (1)(2 , 6)(3 , 5)(4 , 7)(8) 1 5 6 with cyclic type λ • 1 7 2 1 3 Cycle types ⇔ degree 2 1 distributions 8 1 What about σ � and λ � ? σ � = (2 , 3)(1 , 5 , 7 , 8 , 4 , 6) loops around � = faces 2 1 I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 σ • = (1)(2 , 6)(3 , 5)(4 , 7)(8) 1 5 6 with cyclic type λ • 1 7 2 1 3 Cycle types ⇔ degree 2 1 distributions 8 1 What about σ � and λ � ? σ � = (2 , 3)(1 , 5 , 7 , 8 , 4 , 6) loops around � = faces 2 1 But loop around � = concatenate loop around ◦ and • I = S λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
3 critical values, bipartite maps and permutations A loop around a critical value yields a permutation D = S σ ◦ = (1 , 3 , 6)(2 , 5 , 4)(7 , 8) 2 4 with cyclic type λ ◦ 2 1 σ • = (1)(2 , 6)(3 , 5)(4 , 7)(8) 1 5 6 with cyclic type λ • 1 7 2 1 3 Cycle types ⇔ degree 2 1 distributions 8 1 What about σ � and λ � ? σ � = (2 , 3)(1 , 5 , 7 , 8 , 4 , 6) loops around � = faces 2 1 But loop around � = concatenate loop around ◦ and • I = S Proposition: σ ◦ σ • = σ � . λ � = 62 λ • = 2312 λ ◦ = 322 3 critical values 1 regular value with labeled preimages
m + 1 critical values, m -constellations, permutations 1 4 3 2 3 4 2 2 1 3 4 1 2 4 4 1 4 3 1 2 m + 1 critical values 1 2 3 4 1 regular value with labeled preimages
m + 1 critical values, m -constellations, permutations The preimage of the m -star is called a star-constellation. 1 4 Proposition. Planar star-constellations 3 2 3 4 with: 2 2 1 – n labelled m -stars, 3 4 – λ � j faces of degree j , 1 2 – λ ( i ) 4 color i vertices of degree j j 4 are in bijection with minimal transitive 1 factorizations σ 1 · · · σ m = σ � with σ i of cyclic type λ ( i ) . 4 3 1 2 m + 1 critical values 1 2 3 4 1 regular value with labeled preimages
Monodromy, permutations, constellations: a summary Theorem. There is a bijection between • Labelled ramified covering of S of type Λ = ( λ 0 , . . . , λ m ) • Factorizations ( σ 1 · · · σ m = σ 0 ) of type Λ • labelled m -star-constellations of type Λ . D = S ⇔ minimality ⇔ planarity.
Monodromy, permutations, constellations: a summary Theorem. There is a bijection between • Labelled ramified covering of S of type Λ = ( λ 0 , . . . , λ m ) • Factorizations ( σ 1 · · · σ m = σ 0 ) of type Λ • labelled m -star-constellations of type Λ . D = S ⇔ minimality ⇔ planarity. Specializations. — m = 2 : bipartite maps with n edges — m = 2 , λ 0 = 4 n , all faces have degree 4: quadrangulations ⇒ Jean-Fran¸ cois Le Gall’s last year talk at this seminar
Monodromy, permutations, constellations: a summary Theorem. There is a bijection between • Labelled ramified covering of S of type Λ = ( λ 0 , . . . , λ m ) • Factorizations ( σ 1 · · · σ m = σ 0 ) of type Λ • labelled m -star-constellations of type Λ . D = S ⇔ minimality ⇔ planarity. Specializations. — m = 2 : bipartite maps with n edges — m = 2 , λ 0 = 4 n , all faces have degree 4: quadrangulations ⇒ Jean-Fran¸ cois Le Gall’s last year talk at this seminar — for all i ≥ 1 , λ ( i ) = 21 n − 2 : factorizations in transpositions. coverings with only simple branch points
Monodromy, permutations, constellations: a summary Theorem. There is a bijection between • Labelled ramified covering of S of type Λ = ( λ 0 , . . . , λ m ) • Factorizations ( σ 1 · · · σ m = σ 0 ) of type Λ • labelled m -star-constellations of type Λ . D = S ⇔ minimality ⇔ planarity. Specializations. — m = 2 : bipartite maps with n edges — m = 2 , λ 0 = 4 n , all faces have degree 4: quadrangulations ⇒ Jean-Fran¸ cois Le Gall’s last year talk at this seminar — for all i ≥ 1 , λ ( i ) = 21 n − 2 : factorizations in transpositions. coverings with only simple branch points Today’s topic
Simple ramified covers, increasing quadrangulations 6 A ramified cover is simple if its m ramifications have type 21 n − 2 . 1 Then each face of degree 2 on the image has n − 2 preimages that are 3 2 faces of degree 2, and 1 that is a 2 1 quadrangle. 3 4 4 5 5 6 4 1 2 3
Simple ramified covers, increasing quadrangulations 6 A ramified cover is simple if its m ramifications have type 21 n − 2 . 1 Then each face of degree 2 on the image has n − 2 preimages that are 3 2 faces of degree 2, and 1 that is a 2 1 quadrangle. 3 4 Upon contracting multiple edges, 4 only quadrangle remains. 5 5 6 4 1 2 3
Simple ramified covers, increasing quadrangulations 6 A ramified cover is simple if its m ramifications have type 21 n − 2 . 1 Then each face of degree 2 on the image has n − 2 preimages that are 3 2 faces of degree 2, and 1 that is a 2 1 quadrangle. 3 4 Upon contracting multiple edges, 4 only quadrangle remains. 5 Then the faces of the preimage have distinct labels 1 , . . . , m that are increasing in ccw direction around 5 6 black vertices and in cw direction 4 1 around white vertices. 2 3
Simple ramified covers, increasing quadrangulations 6 A ramified cover is simple if its m ramifications have type 21 n − 2 . 1 Then each face of degree 2 on the image has n − 2 preimages that are 3 2 faces of degree 2, and 1 that is a 2 1 quadrangle. 3 4 Upon contracting multiple edges, 4 only quadrangle remains. 5 Then the faces of the preimage have distinct labels 1 , . . . , m that are increasing in ccw direction around 5 6 black vertices and in cw direction 4 1 around white vertices. 2 3 Such a map is called an increasing labelled quadrangulation.
Simple ramified covers, increasing quadrangulations 6 A ramified cover is simple if its m ramifications have type 21 n − 2 . 1 Then each face of degree 2 on the image has n − 2 preimages that are 3 2 faces of degree 2, and 1 that is a 2 1 quadrangle. 3 4 Upon contracting multiple edges, 4 only quadrangle remains. 5 Then the faces of the preimage have distinct labels 1 , . . . , m that are increasing in ccw direction around 5 6 black vertices and in cw direction 4 1 around white vertices. 2 3 Such a map is called an increasing labelled quadrangulation. Theorem. Simple ramified covers of S by itself with m ramifications points are in bijection with increasing labelled quadrangulations with m faces.
R´ esum´ e du 1er ´ episode Compter des classes d’´ equivalence de revˆ etements ramifi´ es ⇔ compter certaines plongements de graphes
Plan de l’expos´ e Revˆ etements ramifi´ es et cartes Cartes et arbres ´ Enum´ eration d’arbres et formule d’Hurwitz Revˆ etements et cartes al´ eatoires
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map.
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree.
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces.
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces.
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree.
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. Proof ?
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. Proof ? Euler’s relation: (#vertices-1)+(#faces-1) = #edges
Planar maps, spanning trees and duality A planar map is a proper embedding of a connected graph on the sphere (considered up to homeomorphisms). From now on, map means rooted planar map. A spanning tree is a subgraph which is a tree and visits every vertices. A tree-rooted map is a map with a spanning tree. The dual map of a map is the map of incidence between faces. The dual map of a tree-rooted map is a tree-rooted map: it is naturally endowed with a dual spanning tree. Proof ? Euler’s relation: (#vertices-1)+(#faces-1) = #edges Proof?
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word uuuduuddddud
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word uuuduuddddud Code of the tree-rooted map = tree decorated by a balanced parenthesis word
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word uuuduuddddud Writing the two codes during the walk: uuuududuuudududddddudd Code of the tree-rooted map = tree decorated by a balanced parenthesis word
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word uuuduuddddud Writing the two codes during the walk: uuuududuuudududddddudd Code of the tree-rooted map = tree decorated by a balanced parenthesis word = shuffle of two balanced parenthesis words
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word uuuduuddddud Writing the two codes during the walk: uuuududuuudududddddudd Code of the tree-rooted map = tree decorated by a balanced parenthesis word = shuffle of two balanced parenthesis words ` 2 n The number of tree rooted planar maps with n edges is P n ´ C i C n − i where i =0 i ` 2 n 1 ´ C n = denotes Catalan numbers, counting balanced parenthesis words. n +1 n
Encoding and counting tree-rooted maps Starting at a root corner, turn around the tree Rooted tree ≡ balanced parenthesis word uduuduuddd Non visited edges ≡ balanced parenthesis word uuuduuddddud Writing the two codes during the walk: uuuududuuudududddddudd Observe that closure edges turn clockwise around the tree. Code of the tree-rooted map = tree decorated by a balanced parenthesis word = shuffle of two balanced parenthesis words ` 2 n The number of tree rooted planar maps with n edges is P n ´ C i C n − i where i =0 i ` 2 n 1 ´ C n = denotes Catalan numbers, counting balanced parenthesis words. n +1 n
but we want rooted (not tree-rooted) maps Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree
but we want rooted (not tree-rooted) maps Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree Then write the code of the primal tree on the chosen canonical tree
but we want rooted (not tree-rooted) maps Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree Then write the code of the primal tree on the chosen canonical tree The map is recovered from the code by closure .
but we want rooted (not tree-rooted) maps Let us recycle the idea used for tree-rooted maps, using a canonical spanning tree Then write the code of the primal tree on the chosen canonical tree The map is recovered from the code by closure . Our code of the map will be a canonical decorated tree Question is How do we choose the canonical spanning tree so that the resulting decorated trees can be described and counted ?
From tree-rooted maps to minimal accessible maps Orient the tree edges away from the root
From tree-rooted maps to minimal accessible maps Orient the tree edges away from the root Orient the other edges couterclockwise around the tree
From tree-rooted maps to minimal accessible maps Orient the tree edges away from the root Orient the other edges couterclockwise around the tree The resulting orientation has no clockwise circuit.
From tree-rooted maps to minimal accessible maps Orient the tree edges away from the root Orient the other edges couterclockwise around the tree The resulting orientation has no clockwise circuit. It is called a minimal orientation (for the order induced by circuit reversal).
From tree-rooted maps to minimal accessible maps Orient the tree edges away from the root Orient the other edges couterclockwise around the tree The resulting orientation has no clockwise circuit. It is called a minimal orientation (for the order induced by circuit reversal). A oriented map is accessible if every vertex can be reach by an oriented path from the root.
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