Rainbow matchings Existence and counting Guillem Perarnau Universitat Polit` ecnica de Catalunya Departament de Matem` atica Aplicada IV 2nd September 2011 Budapest joint work with Oriol Serra
Outline The problem 1 Counting with the Local Lemma 2 Our Approach 3 Random Models 4 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 2 / 19
Outline The problem 1 Counting with the Local Lemma 2 Our Approach 3 Random Models 4 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 3 / 19
Rainbow matchings and Latin transversals → N Edge coloring . C : E ( K n , n ) − Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19
Rainbow matchings and Latin transversals → N Edge coloring . C : E ( K n , n ) − Perfect matching : M = { e i indep } Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19
Rainbow matchings and Latin transversals → N Edge coloring . C : E ( K n , n ) − Perfect matching : M = { e i indep } Rainbow matching : no repeated colors in M . Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19
Rainbow matchings and Latin transversals → N Edge coloring . C : E ( K n , n ) − Integer square matrix A = { a ij } Perfect matching : M = { e i indep } Rainbow matching : no repeated colors in M . 0 1 1 5 4 2 7 2 6 3 B C B C 5 4 2 1 @ A 3 5 3 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19
Rainbow matchings and Latin transversals → N Edge coloring . C : E ( K n , n ) − Integer square matrix A = { a ij } Perfect matching : M = { e i indep } Transversal T σ = { a i σ ( i ) } Rainbow matching : no repeated colors in M . 0 1 1 5 4 2 7 2 6 3 B C B C 5 4 2 1 @ A 3 5 3 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19
Rainbow matchings and Latin transversals → N Edge coloring . C : E ( K n , n ) − Integer square matrix A = { a ij } Perfect matching : M = { e i indep } Transversal T σ = { a i σ ( i ) } Rainbow matching : Latin Transversal : no repeated colors in M . no repeated entries in T σ . 0 1 1 5 4 2 7 2 6 3 B C B C 5 4 2 1 @ A 3 5 3 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 4 / 19
Open problems on Latin squares - Existence Conjecture (Ryser, 1967) Every latin square of odd order admits a latin transversal. Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 5 / 19
Open problems on Latin squares - Existence Conjecture (Ryser, 1967) Every latin square of odd order admits a latin transversal. Conjecture (Brualdi, 1975) Every latin square admits a partial latin transversal of size n − 1. Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 5 / 19
Open problems on Latin squares - Existence Conjecture (Ryser, 1967) Every latin square of odd order admits a latin transversal. Conjecture (Brualdi, 1975) Every latin square admits a partial latin transversal of size n − 1. Theorem (Hatami and Shor, 2008) Every latin square admits a partial latin transversal of size n − O ( log 2 n ) . Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 5 / 19
Open problems on Latin squares - Existence Conjecture (Ryser, 1967) Every latin square of odd order admits a latin transversal. Conjecture (Brualdi, 1975) Every latin square admits a partial latin transversal of size n − 1. Theorem (Hatami and Shor, 2008) Every latin square admits a partial latin transversal of size n − O ( log 2 n ) . Proposition (Erd˝ os and Spencer, 1991) n For every integer matrix, if no entry appears more than 4 e times, then it has a latin transversal. Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 5 / 19
Open problems on Latin squares - Counting Conjecture (Vardi, 1991) Let z n be the number of latin transversals of the cyclic group of order n . Then there exists two constants 0 < c 1 < c 2 < 1 such that c n 1 n ! < z n < c n 2 n ! Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 6 / 19
Open problems on Latin squares - Counting Conjecture (Vardi, 1991) Let z n be the number of latin transversals of the cyclic group of order n . Then there exists two constants 0 < c 1 < c 2 < 1 such that c n 1 n ! < z n < c n 2 n ! Theorem (McKay, McLeod and Wanless, 2006 / Cavenagh and Wan- less, 2010) Let z n be the number of latin transversals of the cyclic group of order n . Then a n < z n < b n √ nn ! where a = 3 . 246 and b = 0 . 614. Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 6 / 19
Outline The problem 1 Counting with the Local Lemma 2 Our Approach 3 Random Models 4 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 7 / 19
Poisson Paradigm A 1 , . . . , A m bad events Pr ( A i ) = p i , m ! \ Pr A i ? i = 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 8 / 19
Poisson Paradigm A 1 , . . . , A m bad events Pr ( A i ) = p i , m ! \ Pr A i ? i = 1 If A i are mutually independent 1 m ! m m \ Y ( 1 − p i ) ∼ e − µ X expected number Pr A i = µ = p i of bad events i = 1 i = 1 i = 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 8 / 19
Poisson Paradigm A 1 , . . . , A m bad events Pr ( A i ) = p i , m ! \ Pr A i ? i = 1 If A i are mutually independent 1 m ! m m \ Y ( 1 − p i ) ∼ e − µ X expected number Pr A i = µ = p i of bad events i = 1 i = 1 i = 1 If µ < 1, by the union bound 2 m ! m \ X Pr A i ≥ 1 − Pr ( A i ) = 1 − µ > 0 i = 1 i = 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 8 / 19
Poisson Paradigm A 1 , . . . , A m bad events Pr ( A i ) = p i , m ! \ Pr A i ? i = 1 If A i are mutually independent 1 m ! m m \ Y ( 1 − p i ) ∼ e − µ X expected number Pr A i = µ = p i of bad events i = 1 i = 1 i = 1 If µ < 1, by the union bound 2 m ! m \ X Pr A i ≥ 1 − Pr ( A i ) = 1 − µ > 0 i = 1 i = 1 Poisson paradigm : If the dependencies among A i are rare. m ! \ = ( 1 + o ( 1 )) e − µ Pr A i i = 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 8 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } E ( H ) = { dependencies among events } Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } E ( H ) = { dependencies among events } , Pr ( A i | T j ∈ S A j ) = Pr ( A i ) Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } E ( H ) = { dependencies among events } , Pr ( A i | T j ∈ S A j ) = Pr ( A i ) ∃ x 1 , . . . , x m ∈ ( 0 , 1 ) such that Y Pr ( A i ) ≤ x i ( 1 − x j ) A j ∈ N ( A i ) Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } E ( H ) = { dependencies among events } , Pr ( A i | T j ∈ S A j ) = Pr ( A i ) ∃ x 1 , . . . , x m ∈ ( 0 , 1 ) such that Y Pr ( A i ) ≤ x i ( 1 − x j ) A j ∈ N ( A i ) Then, m ! \ > 0 Pr A i EXISTENCE i = 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } E ( H ) = { dependencies among events } , Pr ( A i | T j ∈ S A j ) = Pr ( A i ) ∃ x 1 , . . . , x m ∈ ( 0 , 1 ) such that Y Pr ( A i ) ≤ x i ( 1 − x j ) A j ∈ N ( A i ) Then, m ! m \ Y > ( 1 − x i ) Pr A i COUNTING (lower bound) i = 1 i = 1 Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Lov´ asz Local Lemma dependency graph H , V ( H ) = { A 1 , . . . , A m } E ( H ) = { dependencies among events } , Pr ( A i | T j ∈ S A j ) = Pr ( A i ) ∃ x 1 , . . . , x m ∈ ( 0 , 1 ) such that Y Pr ( A i ) ≤ x i ( 1 − x j ) A j ∈ N ( A i ) Then, m ! m \ Y > ( 1 − x i ) Pr A i COUNTING (lower bound) i = 1 i = 1 Lopsided version (Erd˝ os and Spencer, 1991) Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 9 / 19
Upper bound using local Lemma (Lu and Szekely, 2009) ε -near dependency graph H , V ( H ) = { A 1 , . . . , A m } Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 10 / 19
Upper bound using local Lemma (Lu and Szekely, 2009) ε -near dependency graph H , V ( H ) = { A 1 , . . . , A m } Pr ( A i ∩ A j ) = 0 if ( i , j ) ∈ E ( H ) Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 10 / 19
Upper bound using local Lemma (Lu and Szekely, 2009) ε -near dependency graph H , V ( H ) = { A 1 , . . . , A m } Pr ( A i ∩ A j ) = 0 if ( i , j ) ∈ E ( H ) for any S ⊆ [ m ] \ N ( A i ) \ Pr ( A i | A j ) ≥ ( 1 − ε ) Pr ( A i ) j ∈ S Guillem Perarnau MA4-UPC Rainbow matchings: Existence and counting 10 / 19
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