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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

  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

  2. Background Theorem (Ramsey (1928)) There is a function f such that every graph on f ( r ) vertices contains an induced K r or K r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  3. Background Theorem (Ramsey (1928)) There is a function f such that every graph on f ( r ) vertices contains an induced K r or K r . Theorem (Ramsey (1928)) Every giant graph contains a large induced K r or K r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  4. Background Theorem (Ramsey (1928)) Every giant graph contains a large induced K r or K r . Theorem Every giant connected graph contains a large induced K r , K 1 , r , or P r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  5. Background Theorem (Ramsey (1928)) Every giant graph contains a large induced K r or K r . Theorem Every giant connected graph contains a large induced K r , K 1 , r , or P r . Theorem Every giant 2 -connected graph contains a subdivision of K 2 , r or C r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  6. Main Result What about 2-connected induced subgraphs? Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  7. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  8. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. TK + K r TK 2 , r 2 , r DL r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  9. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. TK + K r TK 2 , r 2 , r DL r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  10. Main Result Theorem (A., D.,O. 2019+) Every gargantuan 2 -connected graph contains a large K r , TK 2 , r , TK + 2 , r , or DL r as an induced subgraph. TK + K r TK 2 , r 2 , r DL r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  11. Graphs without a long path Lemma (A.,D.,O.,2019+) Every giant 2-connected graph contains either a long path or an induced TK 2 , r or TK + 2 , r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  12. Graphs without a long path Lemma (A.,D.,O.,2019+) Every giant 2-connected graph contains either a long path or an induced TK 2 , r or TK + 2 , r . TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  13. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  14. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  15. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  16. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  17. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  18. Graphs without a long path TK + TK + TK 2 , r 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  19. Graphs without a long path TK + TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  20. Graphs without a long path TK + TK 2 , r TK 2 , r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  21. Graphs with long path Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced K r , K r , r , or P r . Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  22. Graphs with long path Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced K r , K r , r , or P r . Lemma (A., D.,O. 2019+) Every 2 -connected graph with a really long path contains a large induced K r , K r , r , or long messy ladder. Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  23. Graphs with long path Theorem (Galvin, Rival, Sands) Every graph with a really long path contains a large induced K r , K r , r , or P r . Lemma (A., D.,O. 2019+) Every 2 -connected graph with a really long path contains a large induced K r , K r , r , or long messy ladder. Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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

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

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

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

  28. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  29. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  30. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  31. Long Path to Messy Ladder Q 1 Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  32. Long Path to Messy Ladder Q 1 Q 2 Q 3 P Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  33. Long Path to Messy Ladder Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  34. Long Path to Messy Ladder Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  35. Long Path to Messy Ladder Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  36. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  37. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  38. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  39. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  40. Messy Ladder to Deformed Ladder C r C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  41. Messy Ladder to Deformed Ladder K r TK + C r TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  42. Messy Ladder to Deformed Ladder C r C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  43. 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

  44. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  45. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  46. Messy Ladder to Deformed Ladder C r K r TK + TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

  47. Messy Ladder to Deformed Ladder K r TK + C r TK 2 , r F r 2 , r Sarah Allred*, Guoli Ding, Bogdan Oporowski Unavoidable Induced Subgraphs of Large 2-Connected Graphs

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