List colourings of hypergraphs Andrew Thomason GT2015 24th August - PowerPoint PPT Presentation

List colourings of hypergraphs Andrew Thomason GT2015 24th August 2015

List colourings Let G be a graph or r -uniform hypergraph (edges are r -sets) Let L : V ( G ) → P{ colours } assign a list of colours to each v ∈ G G is k-list-colourable if, whenever ∀ v | L ( v ) | ≥ k , there exists f : V ( G ) → { colours } with f ( v ) ∈ L ( v ), no edge monochromatic The list chromatic number of G is χ l ( G ) = min { k : G is k -list-colourable } Clearly χ l ( G ) ≥ χ ( G ) (make L ( v ) same ∀ v )

χ ℓ can be bigger than χ { 1 , 2 } { 1 , 3 } { 2 , 3 } ✟✟✟✟✟✟✟✟✟✟ ❍ t t t ❅ ❍ � ❅ � ❍ ❅ � ❍ ❅ � ❍ K 3 , 3 not 2-choosable: χ = 2, χ ℓ ≥ 3 ❅ � ❍ � ❅ ❍ � ❅ � ❍ ❅ ❍ � ❅ � ❍ ❅ t t t { 1 , 2 } { 1 , 3 } { 2 , 3 }

χ ℓ can be bigger than χ { 1 , 2 } { 1 , 3 } { 2 , 3 } ❍ ✟✟✟✟✟✟✟✟✟✟ t t t ❅ ❍ � ❅ � ❍ ❅ ❍ � ❅ � ❍ K 3 , 3 not 2-choosable: χ = 2, χ ℓ ≥ 3 ❅ � ❍ � ❅ ❍ � ❅ � ❍ ❅ ❍ � ❅ � ❍ ❅ t t t { 1 , 2 } { 1 , 3 } { 2 , 3 } � 2 k − 1 � More generally, K m , m is not k -choosable if m ≥ k { 1 ,..., k } { ... } { ... } { ... } { k ,..., 2 k − 1 } ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥ ❵ ❳ ✘✘✘✘✘✘✘✘✘✘✘✘ ✘✘✘✘✘✘✘✘✘✘✘✘ ❳ ❵ P ✏✏✏✏✏✏✏✏✏ P ✏✏✏✏✏✏✏✏✏ ✟✟✟✟✟✟ ❍ ❳ ❳ ❍ ✟✟✟✟✟✟ ❵ P P t ❳ ❵ t ❳ t q q q q q t t ❅ ❍ � ❳ ❅ ❵ P � ❳ ❍ P ❅ � ❵ ❳ ❳ ❵ P P ❍ ❳ ❍ ❳ ❵ P P ❳ ❵ ❳ � ❅ ❍ ❅ � ❳ P ❵ ❍ ❳ P ❅ � ❵ ❳ ❳ P ❵ P ❍ ❳ ❍ ❳ ❵ P P ❳ ❳ ❵ � ❅ � ❅ ❍ P ❳ ❍ � ❅ P ❵ ❳ t t t q q q q q t t { 1 ,..., k } { ... } { ... } { ... } { k ,..., 2 k − 1 }

Graphs Theorem (Erd˝ os+Rubin+Taylor 79) χ l ( K d , d ) = (1 + o (1)) log 2 d (upper bound closely tied to “Property B”)

Graphs Theorem (Erd˝ os+Rubin+Taylor 79) χ l ( K d , d ) = (1 + o (1)) log 2 d (upper bound closely tied to “Property B”) Theorem (Alon+Krivelevich 98) whp χ l ( G ( n , n , p )) = (1 + o (1)) log 2 d where G ( n , n , p ) is random bipartite, d = np, d → ∞

Graphs Theorem (Erd˝ os+Rubin+Taylor 79) χ l ( K d , d ) = (1 + o (1)) log 2 d (upper bound closely tied to “Property B”) Theorem (Alon+Krivelevich 98) whp χ l ( G ( n , n , p )) = (1 + o (1)) log 2 d where G ( n , n , p ) is random bipartite, d = np, d → ∞ Conjecture (Alon+Krivelevich 98) For all bipartite G, χ l ( G ) = O (log(∆( G )))

Graphs Theorem (Erd˝ os+Rubin+Taylor 79) χ l ( K d , d ) = (1 + o (1)) log 2 d (upper bound closely tied to “Property B”) Theorem (Alon+Krivelevich 98) whp χ l ( G ( n , n , p )) = (1 + o (1)) log 2 d where G ( n , n , p ) is random bipartite, d = np, d → ∞ Conjecture (Alon+Krivelevich 98) For all bipartite G, χ l ( G ) = O (log(∆( G ))) Theorem (Alon 00) For all graphs G of average degree d, χ l ( G ) ≥ ( 1 2 + o (1)) log 2 d

Bounds χ l ( K d , d ) ≤ log 2 d + 2: Suppose | L ( v ) | ≥ ℓ = log 2 d + 2. For c ∈ { colours } , “forbid” c either on V 1 or on V 2 at random. For each v ∈ V i pick, if poss, f ( v ) ∈ L ( v ) not forbidden on V i . If every v has such a choice, then f colours K d , d . 2 ) ℓ < 1. Expected number of v with no choice is ≤ 2 d ( 1

Bounds χ l ( K d , d ) ≤ log 2 d + 2: Suppose | L ( v ) | ≥ ℓ = log 2 d + 2. For c ∈ { colours } , “forbid” c either on V 1 or on V 2 at random. For each v ∈ V i pick, if poss, f ( v ) ∈ L ( v ) not forbidden on V i . If every v has such a choice, then f colours K d , d . 2 ) ℓ < 1. Expected number of v with no choice is ≤ 2 d ( 1 K ( r ) n , n ,..., n is the complete r -partite r -uniform hypergraph

Bounds χ l ( K d , d ) ≤ log 2 d + 2: Suppose | L ( v ) | ≥ ℓ = log 2 d + 2. For c ∈ { colours } , “forbid” c either on V 1 or on V 2 at random. For each v ∈ V i pick, if poss, f ( v ) ∈ L ( v ) not forbidden on V i . If every v has such a choice, then f colours K d , d . 2 ) ℓ < 1. Expected number of v with no choice is ≤ 2 d ( 1 K ( r ) n , n ,..., n is the complete r -partite r -uniform hypergraph χ l ( K ( r ) 1 n , n ,..., n ) ≤ log r n + 2 = r − 1 log r d + 2: For each c ∈ { colours } , “forbid” c on one of the V i at random. n ) ℓ < 1. Expected number of v with no choice is ≤ rn ( 1

Average degree d and simple hypergraphs Simple or linear hypergraph: | e ∩ f | ≤ 1 for all distinct edges e , f

Average degree d and simple hypergraphs Simple or linear hypergraph: | e ∩ f | ≤ 1 for all distinct edges e , f A Latin square graph is a simple d -regular subgraph of K (3) d , d , d If G Latin square then χ l ( G ) ≤ χ l ( K (3) d , d , d ) ≤ log 3 d + 2.

Average degree d and simple hypergraphs Simple or linear hypergraph: | e ∩ f | ≤ 1 for all distinct edges e , f A Latin square graph is a simple d -regular subgraph of K (3) d , d , d If G Latin square then χ l ( G ) ≤ χ l ( K (3) d , d , d ) ≤ log 3 d + 2. Theorem (Haxell+Pei ’09) If G is a Latin square, d large, then χ l ( G ) = Ω(log d / log log d ) Theorem (Haxell+Verstra¨ ete ’10) � For simple 3-uniform G, ave deg d, χ l ( G ) = Ω( log d / log log d ) Theorem (Alon+Kostochka ’11) For simple r-uniform G, ave deg d, χ l ( G ) = Ω((log d ) 1 / ( r − 1) ))

Hypergraphs Theorem (Saxton+T 12,14) Let G be simple (ie linear) r-uniform d-regular. Then 1 � � χ l ( G ) ≥ r − 1 + o (1) log r d (bounds too for non-regular, non-simple)

Hypergraphs Theorem (Saxton+T 12,14) Let G be simple (ie linear) r-uniform d-regular. Then 1 � � χ l ( G ) ≥ r − 1 + o (1) log r d (bounds too for non-regular, non-simple) Main tool : there’s a collection C ⊂ P V ( G ) of containers such that • for every independent set I , there’s a C ∈ C with I ⊂ C where c = 1 / 4 r 2 • for every C ∈ C , | C | ≤ (1 − c ) | V | • |C| ≤ 2 τ | V | where τ = d − 1 / (2 r − 1)

r -partite hypergraphs 1 (Alon+Krivelevich 98) χ l ( G ( n , n , p )) ∼ log 2 log d , d = np

r -partite hypergraphs 1 (Alon+Krivelevich 98) χ l ( G ( n , n , p )) ∼ log 2 log d , d = np Let G be r -partite r -uniform, average degree d V ( G ) = V 1 ∪ V 2 ∪ · · · ∪ V r , | V i | = n Given X ⊂ V ( G ) write X i = X ∩ V i and let | X j | = max i | X i |

r -partite hypergraphs 1 (Alon+Krivelevich 98) χ l ( G ( n , n , p )) ∼ log 2 log d , d = np Let G be r -partite r -uniform, average degree d V ( G ) = V 1 ∪ V 2 ∪ · · · ∪ V r , | V i | = n Given X ⊂ V ( G ) write X i = X ∩ V i and let | X j | = max i | X i | i � = j | X i | ≤ n r − 1 / d then G [ X ] is 10 3 -degenerate (D r ) if � i � = j | X i | ≥ n r − 1 / d then X is not independent. (N r ) if �

r -partite hypergraphs 1 (Alon+Krivelevich 98) χ l ( G ( n , n , p )) ∼ log 2 log d , d = np Let G be r -partite r -uniform, average degree d V ( G ) = V 1 ∪ V 2 ∪ · · · ∪ V r , | V i | = n Given X ⊂ V ( G ) write X i = X ∩ V i and let | X j | = max i | X i | i � = j | X i | ≤ n r − 1 / d then G [ X ] is 10 3 -degenerate (D r ) if � i � = j | X i | ≥ n r − 1 / d then X is not independent. (N r ) if � Theorem (M´ eroueh+T) If r-partite G satisfies (D r ) and (N r ), in partic if G is random, then χ l ( G ) ∼ g ( r ) log d 1 (Saxton+T 12,14) ∀ simple d -regular G , χ l ( G ) � ( r − 1) log r log d

List colouring Latin squares A latin square is a 3-uniform G with vertices V 1 ⊔ V 2 ⊔ V 3 For every two vertices u , v in different classes, there is exactly one w in the third class such that { u , v , w } is an edge. Thus G is simple and d -regular where d = | V 1 | = | V 2 | = | V 3 | .

List colouring Latin squares A latin square is a 3-uniform G with vertices V 1 ⊔ V 2 ⊔ V 3 For every two vertices u , v in different classes, there is exactly one w in the third class such that { u , v , w } is an edge. Thus G is simple and d -regular where d = | V 1 | = | V 2 | = | V 3 | . χ l ( G ) ≤ 0 · 92 log 3 d : Suppose | L ( v ) | ≥ ℓ = α log 3 d , α = 0 . 92 Let q = 0 . 9083 For each c ∈ { colours } , • with probability q , “forbid” c on one of V 1 , V 2 , V 3 • with probability 1 − q allow c on any V i ( c is “free”) For each v ∈ V i pick, if poss, f ( v ) ∈ L ( v ) forbidden on another V j ; failing that, pick, if poss, a free f ( v ) ∈ L ( v )

List colouring Latin squares A latin square is a 3-uniform G with vertices V 1 ⊔ V 2 ⊔ V 3 For every two vertices u , v in different classes, there is exactly one w in the third class such that { u , v , w } is an edge. Thus G is simple and d -regular where d = | V 1 | = | V 2 | = | V 3 | . χ l ( G ) ≤ 0 · 92 log 3 d : Suppose | L ( v ) | ≥ ℓ = α log 3 d , α = 0 . 92 Let q = 0 . 9083 For each c ∈ { colours } , • with probability q , “forbid” c on one of V 1 , V 2 , V 3 • with probability 1 − q allow c on any V i ( c is “free”) For each v ∈ V i pick, if poss, f ( v ) ∈ L ( v ) forbidden on another V j ; failing that, pick, if poss, a free f ( v ) ∈ L ( v ) E number of v with no choice for f ( v ) is d ( q / 3) ℓ < 1