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Unavoidable Induced Subgraphs of Large 2-Connected Graphs Sarah Allred* Guoli Ding Bogdan Oporowski Louisiana State University 31st Cumberland Conference on Combinatorics, Graph Theory and Computing Background Theorem (Ramsey (1928)) There

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SLIDE 1

Unavoidable Induced Subgraphs of Large 2-Connected Graphs

Sarah Allred* Guoli Ding Bogdan Oporowski

Louisiana State University

31st Cumberland Conference on Combinatorics, Graph Theory and Computing

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SLIDE 2

Background

Theorem (Ramsey (1928)) There is a function f such that every graph on f (r) vertices contains an induced Kr or Kr.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 3

Background

Theorem (Ramsey (1928)) There is a function f such that every graph on f (r) vertices contains an induced Kr or Kr. Theorem (Ramsey (1928)) Every giant graph contains a large induced Kr or Kr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 4

Background

Theorem (Ramsey (1928)) Every giant graph contains a large induced Kr or Kr. Theorem Every giant connected graph contains a large induced Kr, K1,r, or Pr.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 5

Background

Theorem (Ramsey (1928)) Every giant graph contains a large induced Kr or Kr. Theorem Every giant connected graph contains a large induced Kr, K1,r, or Pr. Theorem Every giant 2-connected graph contains a subdivision of K2,r or Cr.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 6

Main Result

What about 2-connected induced subgraphs?

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 7

Main Result

Theorem (A., D.,O. 2019+) Every gargantuan 2-connected graph contains a large Kr, TK2,r, TK +

2,r, or DLr as an induced subgraph.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 8

Main Result

Theorem (A., D.,O. 2019+) Every gargantuan 2-connected graph contains a large Kr, TK2,r, TK +

2,r, or DLr as an induced subgraph.

Kr TK2,r TK +

2,r

DLr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 9

Main Result

Theorem (A., D.,O. 2019+) Every gargantuan 2-connected graph contains a large Kr, TK2,r, TK +

2,r, or DLr as an induced subgraph.

Kr TK2,r TK +

2,r

DLr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 10

Main Result

Theorem (A., D.,O. 2019+) Every gargantuan 2-connected graph contains a large Kr, TK2,r, TK +

2,r, or DLr as an induced subgraph.

Kr TK2,r TK +

2,r

DLr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 11

Graphs without a long path

Lemma (A.,D.,O.,2019+) Every giant 2-connected graph contains either a long path or an induced TK2,r or TK +

2,r.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 12

Graphs without a long path

Lemma (A.,D.,O.,2019+) Every giant 2-connected graph contains either a long path or an induced TK2,r or TK +

2,r.

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 13

Graphs without a long path

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 14

Graphs without a long path

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 15

Graphs without a long path

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 16

Graphs without a long path

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 17

Graphs without a long path

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 18

Graphs without a long path

TK2,r TK +

2,r

TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 19

Graphs without a long path

TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 20

Graphs without a long path

TK2,r TK2,r TK +

2,r

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 21

Graphs with long path

Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced Kr, Kr,r, or Pr.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 22

Graphs with long path

Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced Kr, Kr,r, or Pr. Lemma (A., D.,O. 2019+) Every 2-connected graph with a really long path contains a large induced Kr, Kr,r, or long messy ladder.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 23

Graphs with long path

Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced Kr, Kr,r, or Pr. Lemma (A., D.,O. 2019+) Every 2-connected graph with a really long path contains a large induced Kr, Kr,r, or long messy ladder.

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 24

Tutte Bridges Tutte Bridges

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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Tutte Bridges Tutte Bridges

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 26

Tutte Bridges Tutte Bridges

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 27

Tutte Bridges Tutte Bridges

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 28

Long Path to Messy Ladder

P Q1 Q2 Q3

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 29

Long Path to Messy Ladder

P Q1 Q2 Q3

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 30

Long Path to Messy Ladder

P Q1 Q2 Q3

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 31

Long Path to Messy Ladder

P Q1 Q1 Q2 Q3

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 32

Long Path to Messy Ladder

P Q1 Q2 Q3

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 33

Long Path to Messy Ladder

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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Long Path to Messy Ladder

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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Long Path to Messy Ladder

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 37

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 38

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 39

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 40

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 41

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 42

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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Messy Ladder to Deformed Ladder

After a series of other observations we get...

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 44

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 45

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 46

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 47

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

slide-48
SLIDE 48

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

slide-49
SLIDE 49

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 50

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 51

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

slide-52
SLIDE 52

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

slide-53
SLIDE 53

Messy Ladder to Deformed Ladder

Kr TK2,r TK +

2,r

Fr Cr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 54

Summary

Theorem (A., D.,O. 2019+) Every gargantuan 2-connected graph contains a large Kr, TK2,r, TK +

2,r, or DLr as an induced subgraph.

Kr TK2,r TK +

2,r

DLr

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 55

What’s Next

We are working on the result for infinite graphs

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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SLIDE 56

Thank you for your attention!

Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs