The Rainbow Tur an Problem for Even Cycles Shagnik Das University - - PowerPoint PPT Presentation

The Rainbow Tur´ an Problem for Even Cycles

Shagnik Das

University of California, Los Angeles

Aug 20, 2012 Joint work with Choongbum Lee and Benny Sudakov

Historical Background Rainbow Tur´ an Problem Our Results

Plan

1

Historical Background Tur´ an Problems Colouring Problems

2

Rainbow Tur´ an Problem Definition Motivation Known Results

3

Our Results Summary Warm Up Sketch of Proof

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem

Mantel’s theorem: most fundamental in extremal graph theory Theorem (Mantel, 1907) If a graph G on n vertices has no triangle, then G has at most n2

4

edges.

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem

Mantel’s theorem: most fundamental in extremal graph theory Theorem (Mantel, 1907) If a graph G on n vertices has no triangle, then G has at most n2

4

edges. Tur´ an generalised to cliques of any order Theorem (Tur´ an, 1941) If a graph G on n vertices has no clique of order r, then G has at most

• 1 −

1 r−1 + o(1)

• n2

2 edges.

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem: Extended

Can define Tur´ an numbers of general graphs Definition (Tur´ an numbers) Given any graph H, we define the Tur´ an number ex(n, H) to be the maximum number of edges in an H-free graph on n vertices.

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem: Extended

Can define Tur´ an numbers of general graphs Definition (Tur´ an numbers) Given any graph H, we define the Tur´ an number ex(n, H) to be the maximum number of edges in an H-free graph on n vertices. Erd˝

• s and Stone found asymptotics for all non-bipartite graphs

Theorem (Erd˝

• s-Stone, 1946)

For all graphs H, limn→∞ ex(n, H)/ n

2

• = 1 − (χ(H) − 1)−1.

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem: Open Problems

Tur´ an problem for bipartite graphs generally open

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem: Open Problems

Tur´ an problem for bipartite graphs generally open Particularly interesting is the case of even cycles

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem: Open Problems

Tur´ an problem for bipartite graphs generally open Particularly interesting is the case of even cycles Conjectured upper bound known Theorem (Bondy-Simonovits, 1974) For all k ≥ 2, ex(n, C2k) = O

• n1+ 1

k

• .

Historical Background Rainbow Tur´ an Problem Our Results

Tur´ an’s Theorem: Open Problems

Tur´ an problem for bipartite graphs generally open Particularly interesting is the case of even cycles Conjectured upper bound known Theorem (Bondy-Simonovits, 1974) For all k ≥ 2, ex(n, C2k) = O

• n1+ 1

k

• .

Matching lower bound only known for k = 2, 3, 5

Historical Background Rainbow Tur´ an Problem Our Results

Colouring Problems: Ramsey Theory

Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k, l ≥ 1, there exists R(k, l) such that any red-blue colouring of KR(k,l) contains either a red Kk or a blue Kl.

Historical Background Rainbow Tur´ an Problem Our Results

Colouring Problems: Ramsey Theory

Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k, l ≥ 1, there exists R(k, l) such that any red-blue colouring of KR(k,l) contains either a red Kk or a blue Kl. Determining the Ramsey numbers R(k, l) a widely open problem

Historical Background Rainbow Tur´ an Problem Our Results

Colouring Problems: Ramsey Theory

Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k, l ≥ 1, there exists R(k, l) such that any red-blue colouring of KR(k,l) contains either a red Kk or a blue Kl. Determining the Ramsey numbers R(k, l) a widely open problem

Introduction of the probabilistic method

Historical Background Rainbow Tur´ an Problem Our Results

Colouring Problems: Extensions

Theorem (Erd˝

• s-Rado, 1950)

For every t, there is an n such that every edge-colouring of Kn has a copy of Kt with one of the following canonical colourings: 2 1 3 4

constant

2 1 3 4

rainbow

2 1 3 4

minimum

2 1 3 4

maximum

Historical Background Rainbow Tur´ an Problem Our Results

Colouring Problems: Extensions

Theorem (Erd˝

• s-Rado, 1950)

For every t, there is an n such that every edge-colouring of Kn has a copy of Kt with one of the following canonical colourings: 2 1 3 4

constant

2 1 3 4

rainbow

2 1 3 4

minimum

2 1 3 4

maximum

In particular, for every t there is an n such that every proper edge-colouring of Kn has a rainbow Kt.

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem

First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem

First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete Definition (Rainbow Tur´ an Numbers) Given a graph H, we define the rainbow Tur´ an number ex∗(n, H) to be the maximum number of edges in a properly edge-coloured n-vertex graph with no rainbow copy of H.

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem

First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete Definition (Rainbow Tur´ an Numbers) Given a graph H, we define the rainbow Tur´ an number ex∗(n, H) to be the maximum number of edges in a properly edge-coloured n-vertex graph with no rainbow copy of H. Trivial bound: ex(n, H) ≤ ex∗(n, H)

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem

First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete Definition (Rainbow Tur´ an Numbers) Given a graph H, we define the rainbow Tur´ an number ex∗(n, H) to be the maximum number of edges in a properly edge-coloured n-vertex graph with no rainbow copy of H. Trivial bound: ex(n, H) ≤ ex∗(n, H) Reduction to regular Tur´ an problem

⇒ ex∗(n, H) ≤ ex(n, H) + o(n2)

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Motivation

Definition (Bk-sets) A subset A of an abelian group G is a Bk-set if every g ∈ G has at most one representation of the form g = a1 + a2 + . . . + ak, ai ∈ A.

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Motivation

Definition (Bk-sets) A subset A of an abelian group G is a Bk-set if every g ∈ G has at most one representation of the form g = a1 + a2 + . . . + ak, ai ∈ A. Definition (B∗

k-sets)

A subset A of an abelian group G is a B∗

k-set if there are no two

disjoint k-sets {x1, x2, . . . , xk}, {y1, y2, . . . , yk} ⊂ A with the same sum.

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Motivation

Definition (Bk-sets) A subset A of an abelian group G is a Bk-set if every g ∈ G has at most one representation of the form g = a1 + a2 + . . . + ak, ai ∈ A. Definition (B∗

k-sets)

A subset A of an abelian group G is a B∗

k-set if there are no two

disjoint k-sets {x1, x2, . . . , xk}, {y1, y2, . . . , yk} ⊂ A with the same sum. Example (B∗

k-sets need not be Bk-sets)

Let k = 3, G = Z/6Z, and A = {0, 1, 2, 3, 4, 5}. A is not a B3-set: 0 + 1 + 4 = 0 + 2 + 3 = 0 + 0 + 5. A is a B∗

3-set: the sum of all six elements is odd.

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Connection to Rainbow Cycles

Bipartite Cayley graph construction: B∗

k-sets → rainbow-C2k-free bipartite graphs

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Connection to Rainbow Cycles

Bipartite Cayley graph construction: B∗

k-sets → rainbow-C2k-free bipartite graphs

Construct bipartite graph on X ∪ Y , where X, Y = G Edge (x, y) iff y − x = a ∈ A ⊂ G, colour a

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Connection to Rainbow Cycles

Bipartite Cayley graph construction: B∗

k-sets → rainbow-C2k-free bipartite graphs

Construct bipartite graph on X ∪ Y , where X, Y = G Edge (x, y) iff y − x = a ∈ A ⊂ G, colour a y3 y2 y1 x1 x2 x3 x4 x5 y5 y4

Example with G = Z/5Z, A = {1, 3}

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Connection to Rainbow Cycles

Bipartite Cayley graph construction: B∗

k-sets → rainbow-C2k-free bipartite graphs

Construct bipartite graph on X ∪ Y , where X, Y = G Edge (x, y) iff y − x = a ∈ A ⊂ G, colour a y3 y2 y1 x1 x2 x3 x4 x5 y5 y4

Example with G = Z/5Z, A = {1, 3}

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Connection to Rainbow Cycles

Bipartite Cayley graph construction: B∗

k-sets → rainbow-C2k-free bipartite graphs

Construct bipartite graph on X ∪ Y , where X, Y = G Edge (x, y) iff y − x = a ∈ A ⊂ G, colour a y3 y2 y1 x1 x2 x3 x4 x5 y5 y4

Example with G = Z/5Z, A = {1, 3}

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Connection to Rainbow Cycles

Bipartite Cayley graph construction: B∗

k-sets → rainbow-C2k-free bipartite graphs

Construct bipartite graph on X ∪ Y , where X, Y = G Edge (x, y) iff y − x = a ∈ A ⊂ G, colour a y3 y2 y1 x1 x2 x3 x4 x5 y5 y4

Example with G = Z/5Z, A = {1, 3}

Rainbow C2k x1y1x2y2 . . . xkyk ↔ B = {y1 − x1, . . . , yk − xk} and C = {y1 − x2, y2 − x3, . . . , yk − x1}

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Known Results

A ⊂ G has |A|+k−1

k

• different k-sums

⇒ if |G| = n, A a Bk-set, then |A| = O

• n

1 k

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Known Results

A ⊂ G has |A|+k−1

k

• different k-sums

⇒ if |G| = n, A a Bk-set, then |A| = O

• n

1 k

• Bose and Chowla: Bk-sets of size Ω
• n

1 k

• in G = Z/nZ

⇒ Z/nZ has B∗

k-sets of size Ω

• n

1 k

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Known Results

A ⊂ G has |A|+k−1

k

• different k-sums

⇒ if |G| = n, A a Bk-set, then |A| = O

• n

1 k

• Bose and Chowla: Bk-sets of size Ω
• n

1 k

• in G = Z/nZ

⇒ Z/nZ has B∗

k-sets of size Ω

• n

1 k

• ⇒ ex∗(n, C2k) = Ω
• n1+ 1

k

Historical Background Rainbow Tur´ an Problem Our Results

B∗

k-sets: Known Results

A ⊂ G has |A|+k−1

k

• different k-sums

⇒ if |G| = n, A a Bk-set, then |A| = O

• n

1 k

• Bose and Chowla: Bk-sets of size Ω
• n

1 k

• in G = Z/nZ

⇒ Z/nZ has B∗

k-sets of size Ω

• n

1 k

• ⇒ ex∗(n, C2k) = Ω
• n1+ 1

k

• Ruzsa: O
• n

1 k

• analytic upper bound for B∗

k-sets

Bound on ex∗(n, C2k) would give a combinatorial proof

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Bipartite Graphs

If H has maximum degree s in one part, then ex∗(n, H) = O

• n2− 1

s

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Bipartite Graphs

If H has maximum degree s in one part, then ex∗(n, H) = O

• n2− 1

s

• Gives correct order of magnitude for 4-cycle:

ex∗(n, C4) = O

• n1+ 1

2

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Bipartite Graphs

If H has maximum degree s in one part, then ex∗(n, H) = O

• n2− 1

s

• Gives correct order of magnitude for 4-cycle:

ex∗(n, C4) = O

• n1+ 1

2

• Analysing rainbow paths ⇒ ex∗(n, C6) = O
• n1+ 1

3

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Rainbow Acyclic Graphs

How large can an n-vertex graph without rainbow cycles be?

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Rainbow Acyclic Graphs

How large can an n-vertex graph without rainbow cycles be?

Hypercube construction ⇒ Ω(n log n)

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Rainbow Acyclic Graphs

How large can an n-vertex graph without rainbow cycles be?

Hypercube construction ⇒ Ω(n log n) 6-cycle bound ⇒ O

• n1+ 1

3

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Rainbow Acyclic Graphs

How large can an n-vertex graph without rainbow cycles be?

Hypercube construction ⇒ Ω(n log n) 6-cycle bound ⇒ O

• n1+ 1

3

• Two interesting questions:

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Rainbow Acyclic Graphs

How large can an n-vertex graph without rainbow cycles be?

Hypercube construction ⇒ Ω(n log n) 6-cycle bound ⇒ O

• n1+ 1

3

• Two interesting questions:
• 1. Determine size of rainbow acyclic graphs

Historical Background Rainbow Tur´ an Problem Our Results

Rainbow Tur´ an Problem: Rainbow Acyclic Graphs

How large can an n-vertex graph without rainbow cycles be?

Hypercube construction ⇒ Ω(n log n) 6-cycle bound ⇒ O

• n1+ 1

3

• Two interesting questions:
• 1. Determine size of rainbow acyclic graphs
• 2. Determine asymptotics of ex∗ (n, C2k)

Historical Background Rainbow Tur´ an Problem Our Results

Our Results

Historical Background Rainbow Tur´ an Problem Our Results

Our Results

Theorem (D.-Lee-Sudakov, 2012+) A rainbow acyclic n-vertex graph can have at most n exp

• (log n)

1 2 +η

edges for any η > 0.

Historical Background Rainbow Tur´ an Problem Our Results

Our Results

Theorem (D.-Lee-Sudakov, 2012+) A rainbow acyclic n-vertex graph can have at most n exp

• (log n)

1 2 +η

edges for any η > 0. Theorem (D.-Lee-Sudakov, 2012+) For every integer k ≥ 2, ex∗ (n, C2k) = O

• n1+ (1+εk ) ln k

k

• ,

where εk → 0 as k → ∞.

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up

The black-and-white version of our theorem

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up

The black-and-white version of our theorem Proposition If G on n vertices has more than n1+ε edges, then it has a cycle of length at most 2

ε.

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up

The black-and-white version of our theorem Proposition If G on n vertices has more than n1+ε edges, then it has a cycle of length at most 2

ε.

Proof by induction on n Base case: For n ≤ 2

ε, result follows from existence of cycle

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up

The black-and-white version of our theorem Proposition If G on n vertices has more than n1+ε edges, then it has a cycle of length at most 2

ε.

Proof by induction on n Base case: For n ≤ 2

ε, result follows from existence of cycle

Induction step: May assume minimum degree δ > nε

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up II

v0

Choose an arbitrary vertex v0 ∈ G

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up II

v0 N1 > nε

Expand its neighbourhood N1

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up II

v0 N1 > nε N2 > n2ε

Expand the second neighbourhood N2

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: Warm Up II

v0 N1 > nε N2 > n2ε ... N 1

ε

> n

If no short cycle, then we will eventually exceed n vertices - contradiction

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview

Theorem For every ε > 0, there are constants C(ε) and L(ε) such that any n-vertex graph with at least C(ε)n1+ε edges has a rainbow cycle of length at most L(ε).

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview

Theorem For every ε > 0, there are constants C(ε) and L(ε) such that any n-vertex graph with at least C(ε)n1+ε edges has a rainbow cycle of length at most L(ε). Proof idea: As before, may assume minimum degree δ(G) > C(ε)nε

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview

Theorem For every ε > 0, there are constants C(ε) and L(ε) such that any n-vertex graph with at least C(ε)n1+ε edges has a rainbow cycle of length at most L(ε). Proof idea: As before, may assume minimum degree δ(G) > C(ε)nε Will grow a subtree T out of an arbitrary vertex v0

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

v0

Choose an arbitrary vertex v0 ∈ G

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

v0 L1 nα1

Induction implies the small levels must expand

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

v0 L1 nα1 L2 nα2

Induction implies the small levels must expand

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

v0 L1 nα1 L2 nα2 L3 nα3

Induction implies the small levels must expand

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

v0 L1 nα1 L2 nα2 L3 nα3

We ensure all vertices have a rainbow path back to v0

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

v0 L1 nα1 L2 nα2 L3 nα3

Let Xi,c ⊂ Li be those with colour c in their paths.

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Li nαi

Growing next level: consider unused neighbours of Li

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Li nαi Li+1 nαi+1

Restrict Li+1 to those with between d and 2d neighbours in Li

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Li nαi Li+1 nαi+1 ≈ nε ≈ d

Must have d ≈ nε+αi−αi+1

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Li nαi Li+1 nαi+1

If no short rainbow cycle, then neighbours in Li must share a colour

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Li nαi Li+1 nαi+1

If no short rainbow cycle, then neighbours in Li must share a colour

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Li nαi Li+1 nαi+1

Let Wc be those in Li+1 with many neighbours in Xi,c

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

Restrict to G[Xi,c, Wc]

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

≤ nε ≈ d

If d ≈ nε, short rainbow cycle by induction

Historical Background Rainbow Tur´ an Problem Our Results

Our Proof: A Sketchy Overview II

≤ nε ≈ d

Thus d ≪ nε ⇒ nαi+1 ≫ nαi - desired expansion

Historical Background Rainbow Tur´ an Problem Our Results

Open Problems

Question (Rainbow Acyclicity) We know the largest properly edge-coloured n-vertex graph has at least Ω(n log n) edges, and at most n exp

• (log n)

1 2 +o(1)

edges. Can the gap be narrowed? Question (Rainbow Tur´ an Number for Even Cycles) We have the bounds Ω

• n1+ 1

k

• ≤ ex∗(n, C2k) ≤ O
• n1+ (1+o(1)) ln k

k

• .

Can this gap be narrowed?