Jumps and Non-jumps in q -Multigraphs Steve La Fleur (Joint work - - PowerPoint PPT Presentation

Preliminary Known Results

Jumps and Non-jumps in q-Multigraphs

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

May 14, 2011

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Density

Definition Given a graph G = (V , E) the density of G is defined as d(G) = |E| |V |

2

• Steve La Fleur (Joint work with Paul Horn and Vojtˇ

ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Erd¨

• s-Stone Theorem

Theorem Suppose that ε > 0 and ℓ, m are fixed positive integers. Let G be a graph on n vertices with d(G) ≥ 1 −

1 ℓ−1 + ε. If n > n0(ℓ, m, ε)

then G contains a subgraph isomorphic the complete ℓ partite graph with parts of size m, Tℓ,mℓ.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Erd¨

• s-Stone Theorem

Theorem Suppose that ε > 0 and ℓ, m are fixed positive integers. Let G be a graph on n vertices with d(G) ≥ 1 −

1 ℓ−1 + ε. If n > n0(ℓ, m, ε)

then G contains a subgraph isomorphic the complete ℓ partite graph with parts of size m, Tℓ,mℓ. d(G) > 1 −

1 ℓ−1 and the order of G “large enough”

⇓ G contains a “still large” subgraph with density at least 1 − 1

ℓ

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Let {Gn}∞

n=1 be a sequence of graphs such that |Vn| → ∞ as

n → ∞.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Let {Gn}∞

n=1 be a sequence of graphs such that |Vn| → ∞ as

n → ∞. The maximum density of a subgraph on k vertices is given by σk({Gn}) = max

n

max

V ∈(Vn

k )

d(Gn[V ])

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Upper Density

Definition The upper density of a sequence of graphs {Gn}, denoted as d({Gn}) is given by d({Gn}) = lim

k→∞ σk

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Examples

Consider the sequence of complete, bipartite graphs Gn = Kn,n

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Examples

Consider the sequence of complete, bipartite graphs Gn = Kn,n σk({Kn,n}) =

⌊ k

2 ⌋⌈ k 2 ⌉

(k

2)

= 1

2 + o(1)

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Examples

Consider the sequence of complete, bipartite graphs Gn = Kn,n σk({Kn,n}) =

⌊ k

2 ⌋⌈ k 2 ⌉

(k

2)

= 1

2 + o(1)

d{Kn,n} = 1/2

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Examples

Consider the sequence of complete, bipartite graphs Gn = Kn,n σk({Kn,n}) =

⌊ k

2 ⌋⌈ k 2 ⌉

(k

2)

= 1

2 + o(1)

d{Kn,n} = 1/2 More generally, for a fixed integer ℓ consider the sequence of complete, balanced, ℓ-partite graphs Gn = Tℓ,n d({Gn}) = 1 − 1

ℓ

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Examples

Consider the sequence of complete, bipartite graphs Gn = Kn,n σk({Kn,n}) =

⌊ k

2 ⌋⌈ k 2 ⌉

(k

2)

= 1

2 + o(1)

d{Kn,n} = 1/2 More generally, for a fixed integer ℓ consider the sequence of complete, balanced, ℓ-partite graphs Gn = Tℓ,n d({Gn}) = 1 − 1

ℓ

If {Gn} is a sequence of graphs with d({Gn}) > 1 −

1 ℓ−1,

E-S ⇒ d({Gn}) ≥ d({Tℓ,n}) = 1 − 1

ℓ

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

What is a jump?

Definition A number α is a jump if there exists a constant c = c(α) such that, given any sequence of graphs {Gn}∞

n=1 if d({Gn}) > α then

d({Gn}) ≥ α + c.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

What is a jump?

Definition A number α is a jump if there exists a constant c = c(α) such that, given any sequence of graphs {Gn}∞

n=1 if d({Gn}) > α then

d({Gn}) ≥ α + c. Question Is every α ∈ [0, 1) a jump for simple graphs?

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

What is a jump?

Definition A number α is a jump if there exists a constant c = c(α) such that, given any sequence of graphs {Gn}∞

n=1 if d({Gn}) > α then

d({Gn}) ≥ α + c. Question Is every α ∈ [0, 1) a jump for simple graphs? Answer: Yes! E-S: d({Gn}) > 1 −

1 ℓ−1 ⇒ d({Gn}) ≥ 1 − 1 ℓ

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Harder Questions

Question Is every number in [0, 1) a jump for r-uniform hypergraphs? Erd˝

• s conjectured that the answer was yes and offered $1000 for a

solution.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Harder Questions

Question Is every number in [0, 1) a jump for r-uniform hypergraphs? Erd˝

• s conjectured that the answer was yes and offered $1000 for a

solution. Frankl and R¨

• dl showed that the conjecture was false.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Harder Questions

Question Is every number in [0, 1) a jump for r-uniform hypergraphs? Erd˝

• s conjectured that the answer was yes and offered $1000 for a

solution. Frankl and R¨

• dl showed that the conjecture was false.

Question Is every number a jump for multigraphs of bounded multiplicity? Again Erd˝

• s conjectured that the answer was yes.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results Simple Graphs q-multigraphs

Definitions

Definition A q-multigraph G, is a multigraph with edge multiplicity bounded by q. Density = d(G) =

|E|

(|V |

2 ) ∈ [0, q].

σk = max

n

max

V ∈(Vn

k )

d(G[Vn]) Upper density = d({Gn}) = lim

k→∞ σk

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

[0,1): Everything is a jump!

Proposition Every number in [0, 1) is a jump for q-multigraphs for all values of q.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

[0,1): Everything is a jump!

Proposition Every number in [0, 1) is a jump for q-multigraphs for all values of q. For q = 1 this result follows from the Erd¨

• s-Stone theorem.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

For q ≥ 2

If Gn contains εn2 edges of multiplicity q ≥ 2 E-S implies Gn contains a large complete bipartite graph with edge multiplicity q.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

For q ≥ 2

If Gn contains εn2 edges of multiplicity q ≥ 2 E-S implies Gn contains a large complete bipartite graph with edge multiplicity q. If Gn contains o(n2) edges of multiplicity q, then we can remove them without affecting the upper density.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

For q ≥ 2

If Gn contains εn2 edges of multiplicity q ≥ 2 E-S implies Gn contains a large complete bipartite graph with edge multiplicity q. If Gn contains o(n2) edges of multiplicity q, then we can remove them without affecting the upper density. Conclusion: Edges of higher multiplicity don’t affect the upper density in any non-trivial way on the interval [0, 1).

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

[1,2): Everything is again a jump!!

Proposition Every number in the interval [1, 2) is a jump for all q ≥ 2. Brown, Erd¨

• s and Simonovits together publish a sequence of

papers which imply this proposition in the case q = 2. Sidorenko seperately showed the same result using a different technique.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

For q = 3

If there are εn2 edges of multiplicity three in Gn then d({Gn}) ≥ 3/2

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

For q = 3

If there are εn2 edges of multiplicity three in Gn then d({Gn}) ≥ 3/2 Theorem (Horn,L.,R¨

• dl)

Every number in the interval [0, 2) is a jump for q = 3

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

For q ≥ 4

Again, Gn contains εn2 edges of multiplicity q ≥ 4 implies d({Gn}) ≥ 2 or, Gn contain o(n2) edges of multiplicity q ≥ 4 and we can remove them without changing the upper denisity

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

[3, 4), [4, 5), . . .: Not everything is a jump.

A result of R¨

• dl and Sidorenko gives the following:

Theorem For q ≥ 4, q − 1 is not a jump.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

More non-jumps!

Theorem (Horn, L., R¨

• dl)

Let r ∈ Q ∩ (0, 1]. There exists some positive integer Q such that q − r is not a jump for any q ≥ Q. To show that α is not a jump we find sequences of graphs with upper densities slightly smaller than q − r. We then add an expander graph to these graphs to push the density of this sequence to slightly more than q − r.

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

Open Problems

Question Is every number a jump on [2, 3) for q = 3?

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs

Preliminary Known Results On [0, 1) On [1, 2) For [q-1,q) with q ≥ 4

Thank you!!

Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨

• dl)

Jumps and Non-jumps in q-Multigraphs