Extremalproblems concerningtournaments€¦ · Tournaments • tournament = orientation of a complete graph • two possible 3-vertex subgraphs: C3 and T3 • edge vi → vj for i

Extremal problemsconcerning tournaments

Timothy Chan (Monash)

Andrzej Grzesik (Krakow)

Dan Kral’ (Brno and Warwick)

Jon Noel (Warwick)

May 19, 2019

1

Overview of the talk

• Extremal problems in tournaments

some old and less old results

• Turan type problems in graphs

Turan’s Theorem

Erdos-Rademacher problem

• Turan type problems in tournaments

cycles of length three and four

cycles of arbitrary lengths

2

Tournaments

• tournament = orientation of a complete graph

• two possible 3-vertex subgraphs: C3 and T3

• edge vi → vj for i < j with probability p ∈ [1/2, 1]

• the number of C3 is between 0 and 14

(

n3

)

+O(n2)

• the number of paths u → v → w is at most n3/4

each C3 contains 3 such paths, each T3 one

the number of C3 is at most n3/4−n3/62 = n3

24 +O(n2)

3

Quasirandom tournaments

• When does a tournament look random?

random tournament = orient each edge randomly

• When does a graph look random?

• Thomason, and Chung, Graham and Wilson (1980’s)

density of K2 is p, density of C4 is p4

equivalent subgraph density conditions

equivalent uniform density conditions

equivalent spectral conditions

. . .

4

Quasirandom tournaments

• When does a tournament look random?

• Coregliano, Razborov (2017)

density of T4 is 4!/26(unique minimizer)

density of Tk is k!/2(k

2) for k ≥ 4

• Other tournaments forcing quasirandom?

Coregliano, Parente, Sato (2019)

unique maximizer of a 5-vertex tournament

5





Turan’s Theorem





6

Turan problems

• Maximum edge-density of H-free graph

• Mantel’s Theorem (1907): 12 for H = K3 (Kn

2,n2)

• Turan’s Theorem (1941): ℓ−2ℓ−1 for H = Kℓ (K n

ℓ−1,..., n

ℓ−1)

• Erdos-Stone Theorem (1946): χ(H)−2χ(H)−1

• extremal examples unique up to o(n2) edges

7


• Turan’s Theorem:

edge-density ≤ 1/2 ⇔ minimum triangle density = 0

• What happens if edge-density > 1/2?

• minimum attained by Kn,...,n for edge-density k−1k

• smooth transformation from Kn,n for Kn,n,n,

from Kn,n,n to Kn,n,n,n, etc.

8


K21

K3

1

0

9

History of the problem

• Goodman bound (1959)

d(K3, G) ≥ 2d(K2, G)× ( d(K2, G)− 1/2 )

true for d(K2, G) = k−1k

• Bollobas (1976)

contained in the convex hull

“linear” approximation

• Lovasz and Simonovits (1983)

true for d(K2, G) ∈[

k−1k , k−1

k + εk]

• Fisher (1989)

true for d(K2, G) ∈ [1/2, 2/3]

10

Solution of the problem

• solved by Razborov in 2008

• Flag Algebra Method

calculus for subgraph densities

multiplication of linear combinations

search for true inequalities using SDP

• additional proof idea

differential method (local modifications)

11

Extensions

• Nikiforov (2011)

minimum density of K4

• Reiher (2016)

minimum density of Kr

• Pikhurko and Razborov (2017)

asymptotic structure of extremal graphs

• Liu, Pikhurko and Staden (2017+)

exact structure of extremal graphs

12

Structure of extremal graphs

• Pikhurko and Razborov (2017)

asymptotic structure of extremal graphs

• extremal graphs Kn,...,n,αn

Kn,αn → triangle-free graph on (1 + α)n vertices

• no K1,2 = K1 ∪K2 ⇒ Kn,...,n,αn only

13





Turan’s Theorem





14

Tournaments

• tournament: density parameterized by C3

• analogue of Erdos-Rademacher Problem

minimum density of C4 for a fixed density of C3

• Conjecture of Linial and Morgenstern (2014)

blow-up of a transitive tournament (random inside)

with all but one equal parts and a smaller part

transitive orientation of Kn,...,n,αn, random inside parts

15

Tournaments

• minimum density of C4 for a fixed density of C3

• Conjecture of Linial and Morgenstern (2014)

blow-up of a transitive tournament (random inside)

with all but one equal parts and a smaller part

16

Our results

t(C3, T )

t(C4, T )

0 18

132

172

112

116

1128

1432

17

Approach to the problem

• linear algebra tools

adjacency matrix A ∈ {0, 1}V (G)×V (G)

Tr Ak = number of closed k-walks

• regularity method

approximation by an (n× n)-matrix A

rows and columns ≈ parts in regularity decomposition

Aij ≥ 0 and Aij +Aji = 1 for all i, j ∈ {1, . . . , n}

18

Cases of two and three parts

• non-negative matrix A, s.t. A+ AT = J

• properties of the spectrum of A:

Tr A = λ1 + . . .+ λk = 1/2

Perron–Frobenius ⇒ ∃ρ ∈ R : ρ = λ1 and |λi| ≤ λ1

v∗(A+ AT )v = v∗(λi + λi)v = v∗Jv ≥ 0 ⇒ Re λi ≥ 0

• fix Tr A3 = λ31 + . . .+ λ3

k ∈ [1/36, 1/8]

minimize Tr A4 = λ41 + . . .+ λ4

k

• optimum λ≤k−1 = ρ and λk = 1/2− (k − 1)ρ, k ∈ {2, 3}

19

Case of two parts—structure

• A = (J+B)/2, B is antisymmetric, i.e. B = −BT

A is non-negative and A+ AT = J

• analysis of antisymmetric matrix B

σi and αi for matrix B with∑

i cos2 αi = 1

B = UT

0 σ1 0 0

−σ1 0 0 0

0 0 0 σ2

0 0 −σ2 0

U

20



A is non-negative and A+ AT = J

• analysis of antisymmetric matrix B


i cos2 αi = 1

• Tr A3 ≈ Tr J3 + Tr JB2 =∑

i σ2i cos

2 αi

Tr A4 ≈ Tr J4 + Tr J2B2 + Tr B4 ≈ Tr JB2 +∑

i σ4i

• optimum for α1 = 0, α≥2 = π/2 and σ≥2 = 0

21




i cos2 αi = 1

optimum for α1 = 0, α≥2 = π/2 and σ≥2 = 0

• assign pv ∈ [0, 1/2] to each vertex v

orient from v to w with probability 1/2 + (pv − pw)

• conjectured construction: pv ∈ {0, 1/2}

22

Maximum density of cycles

• work in progress with Grzesik, Lovasz Jr. and Volec

• What is maximum density of cycles of length k?

k ≡ 1 mod 4 ⇔ regular tournament

k ≡ 2 mod 4 ⇔ quasirandom tournament

k ≡ 3 mod 4 ⇔ regular tournament

k ≡ 4 mod 4 ⇔ ????

• “cyclic” tournament for k = 4 and k = 8

23

Thank you for your attention!

24

Documents

Extremalproblems concerningtournaments€¦ · Tournaments • tournament = orientation of a complete graph • two possible 3-vertex subgraphs: C3 and T3 • edge vi → vj for i