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 χl(G) = min{k : G is k-list-colourable} Clearly χl(G) ≥ χ(G) (make L(v) same ∀v)

χℓ can be bigger than χ

K3,3 not 2-choosable: χ = 2, χℓ ≥ 3 {1, 2} {1, 3} {2, 3} {1, 2} {1, 3} {2, 3}

• ✟✟✟✟✟✟✟✟✟✟

❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ t t t t t t

χℓ can be bigger than χ

K3,3 not 2-choosable: χ = 2, χℓ ≥ 3 {1, 2} {1, 3} {2, 3} {1, 2} {1, 3} {2, 3}

• ✟✟✟✟✟✟✟✟✟✟

❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ t t t t t t

More generally, Km,m is not k-choosable if m ≥ 2k−1

k

• {1,...,k}

{...} {...} {...} {k,...,2k−1} {1,...,k} {...} {...} {...} {k,...,2k−1}

• ✟✟✟✟✟✟

✘✘✘✘✘✘✘✘✘✘✘✘ ✥✥✥✥✥✥✥✥✥✥✥✥✥✥✥ ❅ ❅ ❅

• ✏✏✏✏✏✏✏✏✏

✘✘✘✘✘✘✘✘✘✘✘✘ ❍ ❍ ❍ ❍ ❍ ❍ ❅ ❅ ❅ ✟✟✟✟✟✟ ✏✏✏✏✏✏✏✏✏ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ P P P P P P P P P ❍ ❍ ❍ ❍ ❍ ❍

• ❵

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❳ P P P P P P P P P ❅ ❅ ❅ t t t q q q q q t t t t t q q q q q t t

Graphs

Theorem (Erd˝

• s+Rubin+Taylor 79)

χl(Kd,d) = (1 + o(1)) log2 d (upper bound closely tied to “Property B”)

Graphs

Theorem (Erd˝

• s+Rubin+Taylor 79)

χl(Kd,d) = (1 + o(1)) log2 d (upper bound closely tied to “Property B”)

Theorem (Alon+Krivelevich 98)

whp χl(G(n, n, p)) = (1 + o(1)) log2 d where G(n, n, p) is random bipartite, d = np, d → ∞

Graphs

Theorem (Erd˝

• s+Rubin+Taylor 79)

χl(Kd,d) = (1 + o(1)) log2 d (upper bound closely tied to “Property B”)

Theorem (Alon+Krivelevich 98)

whp χl(G(n, n, p)) = (1 + o(1)) log2 d where G(n, n, p) is random bipartite, d = np, d → ∞

Conjecture (Alon+Krivelevich 98)

For all bipartite G, χl(G) = O(log(∆(G)))

Graphs

Theorem (Erd˝

• s+Rubin+Taylor 79)

χl(Kd,d) = (1 + o(1)) log2 d (upper bound closely tied to “Property B”)

Theorem (Alon+Krivelevich 98)

whp χl(G(n, n, p)) = (1 + o(1)) log2 d where G(n, n, p) is random bipartite, d = np, d → ∞

Conjecture (Alon+Krivelevich 98)

For all bipartite G, χl(G) = O(log(∆(G)))

Theorem (Alon 00)

For all graphs G of average degree d, χl(G) ≥ ( 1

2 + o(1)) log2 d

Bounds

χl(Kd,d) ≤ log2 d + 2: Suppose |L(v)| ≥ ℓ = log2 d + 2. For c ∈ {colours}, “forbid” c either on V1 or on V2 at random. For each v ∈ Vi pick, if poss, f (v) ∈ L(v) not forbidden on Vi. If every v has such a choice, then f colours Kd,d. Expected number of v with no choice is ≤ 2d( 1

2)ℓ < 1.

Bounds

χl(Kd,d) ≤ log2 d + 2: Suppose |L(v)| ≥ ℓ = log2 d + 2. For c ∈ {colours}, “forbid” c either on V1 or on V2 at random. For each v ∈ Vi pick, if poss, f (v) ∈ L(v) not forbidden on Vi. If every v has such a choice, then f colours Kd,d. Expected number of v with no choice is ≤ 2d( 1

2)ℓ < 1.

K (r)

n,n,...,n is the complete r-partite r-uniform hypergraph

Bounds

χl(Kd,d) ≤ log2 d + 2: Suppose |L(v)| ≥ ℓ = log2 d + 2. For c ∈ {colours}, “forbid” c either on V1 or on V2 at random. For each v ∈ Vi pick, if poss, f (v) ∈ L(v) not forbidden on Vi. If every v has such a choice, then f colours Kd,d. Expected number of v with no choice is ≤ 2d( 1

2)ℓ < 1.

K (r)

n,n,...,n is the complete r-partite r-uniform hypergraph

χl(K (r)

n,n,...,n) ≤ logr n + 2 = 1 r−1 logr d + 2:

For each c ∈ {colours}, “forbid” c on one of the Vi at random. Expected number of v with no choice is ≤ rn( 1

n)ℓ < 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 χl(G) ≤ χl(K (3)

d,d,d) ≤ log3 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 χl(G) ≤ χl(K (3)

d,d,d) ≤ log3 d + 2.

Theorem (Haxell+Pei ’09)

If G is a Latin square, d large, then χl(G) = Ω(log d/ log log d)

Theorem (Haxell+Verstra¨ ete ’10)

For simple 3-uniform G, ave deg d, χl(G) = Ω(

• log d/ log log d)

Theorem (Alon+Kostochka ’11)

For simple r-uniform G, ave deg d, χl(G) = Ω((log d)1/(r−1)))

Hypergraphs

Theorem (Saxton+T 12,14)

Let G be simple (ie linear) r-uniform d-regular. Then χl(G) ≥

• 1

r − 1 + o(1)

• logr 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 χl(G) ≥

• 1

r − 1 + o(1)

• logr d

(bounds too for non-regular, non-simple) Main tool: there’s a collection C ⊂ PV (G) of containers such that

• for every independent set I, there’s a C ∈ C with I ⊂ C
• for every C ∈ C, |C| ≤ (1 − c)|V |

where c = 1/4r2

• |C| ≤ 2τ|V |

where τ = d−1/(2r−1)

r-partite hypergraphs

(Alon+Krivelevich 98) χl(G(n, n, p)) ∼

1 log 2 log d, d = np

r-partite hypergraphs

(Alon+Krivelevich 98) χl(G(n, n, p)) ∼

1 log 2 log d, d = np

Let G be r-partite r-uniform, average degree d V (G) = V1 ∪ V2 ∪ · · · ∪ Vr, |Vi| = n Given X ⊂ V (G) write Xi = X ∩ Vi and let |Xj| = maxi |Xi|

r-partite hypergraphs

(Alon+Krivelevich 98) χl(G(n, n, p)) ∼

1 log 2 log d, d = np

Let G be r-partite r-uniform, average degree d V (G) = V1 ∪ V2 ∪ · · · ∪ Vr, |Vi| = n Given X ⊂ V (G) write Xi = X ∩ Vi and let |Xj| = maxi |Xi| (Dr) if

i=j |Xi| ≤ nr−1/d then G[X] is 103-degenerate

(Nr) if

i=j |Xi| ≥ nr−1/d then X is not independent.

r-partite hypergraphs

(Alon+Krivelevich 98) χl(G(n, n, p)) ∼

1 log 2 log d, d = np

Let G be r-partite r-uniform, average degree d V (G) = V1 ∪ V2 ∪ · · · ∪ Vr, |Vi| = n Given X ⊂ V (G) write Xi = X ∩ Vi and let |Xj| = maxi |Xi| (Dr) if

i=j |Xi| ≤ nr−1/d then G[X] is 103-degenerate

(Nr) if

i=j |Xi| ≥ nr−1/d then X is not independent.

Theorem (M´ eroueh+T)

If r-partite G satisfies (Dr) and (Nr), in partic if G is random, then χl(G) ∼ g(r) log d (Saxton+T 12,14) ∀ simple d-regular G, χl(G)

1 (r−1) log r log d

List colouring Latin squares

A latin square is a 3-uniform G with vertices V1 ⊔ V2 ⊔ V3 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 = |V1| = |V2| = |V3|.

List colouring Latin squares

A latin square is a 3-uniform G with vertices V1 ⊔ V2 ⊔ V3 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 = |V1| = |V2| = |V3|. χl(G) ≤ 0 · 92 log3 d: Suppose |L(v)| ≥ ℓ = α log3 d, α = 0.92 Let q = 0.9083 For each c ∈ {colours},

• with probability q, “forbid” c on one of V1, V2, V3
• with probability 1 − q allow c on any Vi (c is “free”)

For each v ∈ Vi pick, if poss, f (v) ∈ L(v) forbidden on another Vj; 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 V1 ⊔ V2 ⊔ V3 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 = |V1| = |V2| = |V3|. χl(G) ≤ 0 · 92 log3 d: Suppose |L(v)| ≥ ℓ = α log3 d, α = 0.92 Let q = 0.9083 For each c ∈ {colours},

• with probability q, “forbid” c on one of V1, V2, V3
• with probability 1 − q allow c on any Vi (c is “free”)

For each v ∈ Vi pick, if poss, f (v) ∈ L(v) forbidden on another Vj; 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

List colouring Latin squares

A latin square is a 3-uniform G with vertices V1 ⊔ V2 ⊔ V3 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 = |V1| = |V2| = |V3|. χl(G) ≤ 0 · 92 log3 d: Suppose |L(v)| ≥ ℓ = α log3 d, α = 0.92 Let q = 0.9083 For each c ∈ {colours},

• with probability q, “forbid” c on one of V1, V2, V3
• with probability 1 − q allow c on any Vi (c is “free”)

For each v ∈ Vi pick, if poss, f (v) ∈ L(v) forbidden on another Vj; 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 E number of monochromatic edges is ≤ d2(1 − q)ℓ < 1

Preference orders

A preference order on [m] is a collection of r total orders on [m]

Preference orders

A preference order on [m] is a collection of r total orders on [m] Example: r = 2 m = 2k

position

2k 1 1 2k − 1 2 1 − 1/m . . . . . . . . . k + 1 k − 1 1/2 + 1/m k k 1/2 k − 1 k + 1 1/2 − 1/m . . . . . . . . . 2 2k − 1 2/m 1 2k 1/m

Preference orders

A preference order on [m] is a collection of r total orders on [m] Example: r = 3 m = 3k

position

3k k 2k 1 3k − 1 k − 1 2k − 1 1 − 1/m . . . . . . . . . . . . 2k + 1 1 k + 1 2/3 + 1/m 2k 3k k 2/3 2k − 1 3k − 1 k − 1 2/3 − 1/m . . . . . . . . . . . . k + 1 2k + 1 1 1/3 + 1/m 1 k + 1 2k + 1 1/3 2 k + 2 2k + 2 1/3 − 1/m . . . . . . . . . . . . k 2k 3k 1/m

Preference orders - naive algorithm

Algorithm: randomly label the {colours} 1, . . . , 3k. ∗ v ∈ Vi chooses f (v) ∈ L(v) highest in the preferred order for Vi Idea: suppose edge {v1, v2, v3} is monochromatic with colour c. Then c is in worst third for (say) v1 and middle third for (say) v3. Then L(v1) ∩ L(v3) = {c} else both would not choose c. If c is relative height 1/3 − x, 0 ≤ x ≤ 1/3, in the V1 order then it is height 1/3 + x in V3 order. Hence Pr[edge is monochromatic] ≤ ( 1

3 − x)ℓ( 1 3 + x)ℓ ≤ ( 1 9)ℓ.

Preference orders - naive algorithm

Algorithm: randomly label the {colours} 1, . . . , 3k. ∗ v ∈ Vi chooses f (v) ∈ L(v) highest in the preferred order for Vi Idea: suppose edge {v1, v2, v3} is monochromatic with colour c. Then c is in worst third for (say) v1 and middle third for (say) v3. Then L(v1) ∩ L(v3) = {c} else both would not choose c. If c is relative height 1/3 − x, 0 ≤ x ≤ 1/3, in the V1 order then it is height 1/3 + x in V3 order. Hence Pr[edge is monochromatic] ≤ ( 1

3 − x)ℓ( 1 3 + x)ℓ ≤ ( 1 9)ℓ.

E number of monochromatic edges is ≤ d2( 1

9)ℓ < 1 if ℓ > log3 d

Preference orders - naive algorithm

Algorithm: randomly label the {colours} 1, . . . , 3k. ∗ v ∈ Vi chooses f (v) ∈ L(v) highest in the preferred order for Vi Idea: suppose edge {v1, v2, v3} is monochromatic with colour c. Then c is in worst third for (say) v1 and middle third for (say) v3. Then L(v1) ∩ L(v3) = {c} else both would not choose c. If c is relative height 1/3 − x, 0 ≤ x ≤ 1/3, in the V1 order then it is height 1/3 + x in V3 order. Hence Pr[edge is monochromatic] ≤ ( 1

3 − x)ℓ( 1 3 + x)ℓ ≤ ( 1 9)ℓ.

E number of monochromatic edges is ≤ d2( 1

9)ℓ < 1 if ℓ > log3 d

But bring back fixed colours and use preferences on free colours gives χl(G) ≤ 0.77 log3 d

Preference orders - values

The value of a preference order is max

j∈[m]

product of lowest r − 1 positions of j

Preference orders - values

The value of a preference order is max

j∈[m]

product of lowest r − 1 positions of j 2k 1 2k − 1 2 . . . . . . k + 1 k − 1 k k value maxx≤1/2 x = 1/2 k − 1 k + 1 . . . . . . 2 2k − 1 1 2k

Preference orders - values

The value of a preference order is max

j∈[m]

product of lowest r − 1 positions of j 3k k 2k 3k − 1 k − 1 2k − 1 . . . . . . . . . 2k + 1 1 k + 1 2k 3k k 2k − 1 3k − 1 k − 1 . . . . . . . . . value maxx (1/3 + x)(1/3 − x) = 1/9 k + 1 2k + 1 1 1 k + 1 2k + 1 2 k + 2 2k + 2 . . . . . . . . . k 2k 3k

Preference orders - f (r) and g(r)

M´ eroueh+T ∀ r-partite G, (Dr) & (Nr) ⇒ χl(G) ∼ g(r) log d (Saxton+T 12,14) ∀ simple d-regular G, χl(G)

1 (r−1) log r log d

Preference orders - f (r) and g(r)

M´ eroueh+T ∀ r-partite G, (Dr) & (Nr) ⇒ χl(G) ∼ g(r) log d (Saxton+T 12,14) ∀ simple d-regular G, χl(G)

1 (r−1) log r log d

Define f (r) to be the minimum value of all pref orders (as m → ∞) f (2) = 1/2 f (3) = 1/9 1/64 ≤ f (4) ≤ 1/32 f (r) =?

Preference orders - f (r) and g(r)

M´ eroueh+T ∀ r-partite G, (Dr) & (Nr) ⇒ χl(G) ∼ g(r) log d (Saxton+T 12,14) ∀ simple d-regular G, χl(G)

1 (r−1) log r log d

Define f (r) to be the minimum value of all pref orders (as m → ∞) f (2) = 1/2 f (3) = 1/9 1/64 ≤ f (4) ≤ 1/32 f (r) =? Define g(r) =

1 − log f (r)

which means f (r)g(r) = 1

e

Preference orders - f (r) and g(r)

M´ eroueh+T ∀ r-partite G, (Dr) & (Nr) ⇒ χl(G) ∼ g(r) log d (Saxton+T 12,14) ∀ simple d-regular G, χl(G)

1 (r−1) log r log d

Define f (r) to be the minimum value of all pref orders (as m → ∞) f (2) = 1/2 f (3) = 1/9 1/64 ≤ f (4) ≤ 1/32 f (r) =? Define g(r) =

1 − log f (r)

which means f (r)g(r) = 1

e

g(2) =

1 log 2

g(3) =

1 2 log 3

show container argument tight r ≤ 3

Preference orders - f (r) and g(r)

M´ eroueh+T ∀ r-partite G, (Dr) & (Nr) ⇒ χl(G) ∼ g(r) log d (Saxton+T 12,14) ∀ simple d-regular G, χl(G)

1 (r−1) log r log d

Define f (r) to be the minimum value of all pref orders (as m → ∞) f (2) = 1/2 f (3) = 1/9 1/64 ≤ f (4) ≤ 1/32 f (r) =? Define g(r) =

1 − log f (r)

which means f (r)g(r) = 1

e

g(2) =

1 log 2

g(3) =

1 2 log 3

show container argument tight r ≤ 3 If f (4) = 1/64, g(4) =

1 3 log 4

then containers tight r = 4

Proof of upper bound —

bells

algorithm

G satisfies (Dr): given lists size ℓ = (1 + ǫ)g(r) log d, colour G

Proof of upper bound —

bells

algorithm

G satisfies (Dr): given lists size ℓ = (1 + ǫ)g(r) log d, colour G Split colours randomly into m = ǫℓ/103 blocks, labelled 1, . . . , m Take a preference order with value f (r) on the blocks

Proof of upper bound —

bells

algorithm

G satisfies (Dr): given lists size ℓ = (1 + ǫ)g(r) log d, colour G Split colours randomly into m = ǫℓ/103 blocks, labelled 1, . . . , m Take a preference order with value f (r) on the blocks Given v ∈ Vi, assign v to highest block B in i’th order in which v’s list has ≥ 103 colours ; ≥ (1 − ǫ)ℓ of list is in such blocks x1 x2 x3 x4 B B B B

Proof of upper bound —

bells

algorithm

G satisfies (Dr): given lists size ℓ = (1 + ǫ)g(r) log d, colour G Split colours randomly into m = ǫℓ/103 blocks, labelled 1, . . . , m Take a preference order with value f (r) on the blocks Given v ∈ Vi, assign v to highest block B in i’th order in which v’s list has ≥ 103 colours ; ≥ (1 − ǫ)ℓ of list is in such blocks x1 x2 x3 x4 B B B B Pr{v ∈ Vi assigned to B position xi} ≤ x(1−ǫ)ℓ

i

≤ xg(r) log d

i

Proof of upper bound —

bells

algorithm

G satisfies (Dr): given lists size ℓ = (1 + ǫ)g(r) log d, colour G Split colours randomly into m = ǫℓ/103 blocks, labelled 1, . . . , m Take a preference order with value f (r) on the blocks Given v ∈ Vi, assign v to highest block B in i’th order in which v’s list has ≥ 103 colours ; ≥ (1 − ǫ)ℓ of list is in such blocks x1 x2 x3 x4 B B B B Pr{v ∈ Vi assigned to B position xi} ≤ x(1−ǫ)ℓ

i

≤ xg(r) log d

i

Let X = vertices assigned to B. Then |Xi| ≤ xg(r) log d

i

n so |X1||X3||X4| ≤ (x1x3x4)g(r) log dn3 ≤ f (r)g(r) log dn3 = n3/d By (Dr) X is 103-degenerate so can colour with colours in B

Proof of lower bound

G satisfies (Nr): if lists size ℓ chosen at random from a pallette of t colours, t ≈ log2 d, then whp there’s no colouring if ℓ < g(r) log d

Proof of lower bound

G satisfies (Nr): if lists size ℓ chosen at random from a pallette of t colours, t ≈ log2 d, then whp there’s no colouring if ℓ < g(r) log d Suppose such a colouring exists. Form a preference order with [m] = [t]: i’th order by how many vertices in Vi get each colour

Proof of lower bound

G satisfies (Nr): if lists size ℓ chosen at random from a pallette of t colours, t ≈ log2 d, then whp there’s no colouring if ℓ < g(r) log d Suppose such a colouring exists. Form a preference order with [m] = [t]: i’th order by how many vertices in Vi get each colour x1 x2 x3 x4

t t t t

Some colour (green, say) satisfies x1x3x4 ≥ f (4).

Proof of lower bound

G satisfies (Nr): if lists size ℓ chosen at random from a pallette of t colours, t ≈ log2 d, then whp there’s no colouring if ℓ < g(r) log d Suppose such a colouring exists. Form a preference order with [m] = [t]: i’th order by how many vertices in Vi get each colour x1 x2 x3 x4

t t t t

Some colour (green, say) satisfies x1x3x4 ≥ f (4). Pr{list ⊂ cols below green in i’th} = xℓ

i ⇒ these cols on xℓ i n in Vi

Proof of lower bound

G satisfies (Nr): if lists size ℓ chosen at random from a pallette of t colours, t ≈ log2 d, then whp there’s no colouring if ℓ < g(r) log d Suppose such a colouring exists. Form a preference order with [m] = [t]: i’th order by how many vertices in Vi get each colour x1 x2 x3 x4

t t t t

Some colour (green, say) satisfies x1x3x4 ≥ f (4). Pr{list ⊂ cols below green in i’th} = xℓ

i ⇒ these cols on xℓ i n in Vi

If X = green verts then |X1||X3||X4| ≥ xℓ

1xℓ 3xℓ 4n3/t3 ≥ f (4)ℓn3/t3

Proof of lower bound

G satisfies (Nr): if lists size ℓ chosen at random from a pallette of t colours, t ≈ log2 d, then whp there’s no colouring if ℓ < g(r) log d Suppose such a colouring exists. Form a preference order with [m] = [t]: i’th order by how many vertices in Vi get each colour x1 x2 x3 x4

t t t t

Some colour (green, say) satisfies x1x3x4 ≥ f (4). Pr{list ⊂ cols below green in i’th} = xℓ

i ⇒ these cols on xℓ i n in Vi

If X = green verts then |X1||X3||X4| ≥ xℓ

1xℓ 3xℓ 4n3/t3 ≥ f (4)ℓn3/t3

So |X1||X3||X4| > f (4)g(4) log dn3 = n3/d. By (N4) X not indept #

So in fact how big is f (r)?

1 38.13 < f (4) < 1 38.12 all numbers appear once in top quarter and once in bottom quarter some twice here

✲

and once here

✲

rest once here and weirdly here with constant prods

✲ ✲

* * * * * * * * * * * * * * * * So containers not tight for r = 4 And r−1

re

r−1 ≤ f (r) ≤ (r−1)!

rr−1

so g(r) ∼ 1

r c.f. container 1 (r−1) log r