E xtremal S ubgraphs of R andom G raphs

1. Extremal Subgraphs ofRandom Graphs Konstantinos Panagiotou Institute ofTheoretical Computer Science ETH Zürich (joint work with Graham Brightwell and Angelika Steger)

2. Maximum bipartite subgraph Motivation Maximum triangle-free subgraph

3. Maximum bipartite subgraph But… Maximum triangle-free subgraph

4. This Talk Probabilistic Viewpoint

5. Random Graph . set of maximum triangle-free subgraphs set of maximum bipartite subgraphs size of a maximum triangle-free subgraph size of a maximum bipartite subgraph Questions: • ? • If so, do we also have ?

6. Random Graph . here: [Mantel 1907] we know from random graph theory: is aas. a forest hence: aas. ?

7. Easy to see: Necessary Condition: no edge within the larger class can be added (without creating a triangle) Pr[edge can be added] Pr[no edge can be added] If then

8. Random Graph . ?

9. A Related Problem Erdös, Kleitman, Rothschild (1976) Almost all large triangle-freegraphs are bipartite. randomtriangle-freegraph on verticeswithedges Prömel, Steger (1996), Steger (2005), Osthus, Prömel, Taraz (2003):

10. Pictures say more… Could a similar picture be true for the random graph ?

11. - What we know… Babai, Simonovits, Spencer (1990) ? Question (Babai, Simonovits, Spencer): Do we have for all constant ? What if ?

12. In fact, the theorem is true for all ³2 if we replace “triangle-free” with “ -free” and “bipartite” with “ -partite”. Our result Brightwell, P., Steger If then

13. Szemerédi’s regularity lemma Only “few” rededges Properties of Proof pipeline 1 2 Even less rededges 3 Properties of maximum cuts

14. Proof: Part . 1 Szemerédi’s Regularity Lemma (sparse case) Kohayakawa (1997) Rödl (2003) Pair is regular anddense (Probabilistic) Embedding Lemma Gerke, Marciniszyn, Steger (2005) Contradiction! (Embedding Lemma)

15. Szemeredi’s regularity lemma Only “few” rededges Properties of Proof pipeline 1 2 Even less rededges 3 Properties of maximum cuts

16. Proof: Part . 2 • Start with a „best“ bipartition • „best“: the number of red edges is minimized • Main idea: if we can exhibit more „missing“ edges than red edges, then we achieve a contradiction „missing“

17. Proof: Part - setup. 2 Vertices which do not fulfill the “common neighb.” property “Too many” missing edges: • All „exceptional sets“ are small • „Remainder“ is very sparse • Bootstrapping

18. Szemeredi’s regularity lemma Only “few” rededges Properties of Proof pipeline 1 2 Even less rededges 3 Properties of maximum cuts

19. Proof: Part . 3 How does a maximum triangle-free subgraph - of look like? is a bipartition red edges Observe: has to contain all edges of between except possibly some edges has to be a partition that is within to an optimum partition

20. Proof: Part - continued 3 Howlikelyisitthatonecanadd a singlerededge after removing atmostedgesfrom (withoutcreating a triangle)? • There are disjoint triangles with and ... • Hence Need: number of partitions to be considered

21. On the number of optimal bipartitions • Let and be two bipartitions. Then • Theorem. LetThe probability that has two optimal bipartitions and with is .

22. Sketch of proof • Let denote the size of max-cut • Consider the expected change Two points of view • Delete edge from • Add edge to bring desired probability into play

23. Deleting edges • If we delete edges, the max-cut decreases by at most the number of edges which are removed from all max-cuts • Upper bound: consider a fixed max-cut • Hence

24. Adding edges • Fix a max-cut in • When adding edges, one of two cases occurs: • remains (one of) the max-cuts • A different max-cut “overtakes” • Hence and

25. Adding edges (II) • Increase of : number of edges added that go across • Depends on the number of non-edges across in • Hence

26. Adding edges (III) • Suppose “overtakes” • Also: • If the event occurs, then there were added more edges to than to • It turns out • Putting the upper & lower bounds together yields the theorem

27. Szemeredi’s regularity lemma Only “few” rededges Properties of Proof pipeline 1 2 Even less rededges 3 Properties of maximum cuts

28. Open Problem ?

29. Summary • We showed that for sufficiently dense random graphs the maximum triangle-free and maximum bipartite subgraphs are identical. • On the way we obtained information about the structure of the max-cut in random graphs. • Our results generalize to arbitrary (constant-size) cliques.

30. Questions?

31. Warm-up for Part . 2 Properties of random graphs (for ): • degree property • degree property for subsets • density property (next three slides)

32. For all Degree Property • has the following property (aas): we have and

33. For all and all except of vertices Degree Property for Subsets • has the following property (aas): we have and

34. with we have “Density” property • has the following property (aas): For all ,

35. Vertices which do not fulfill the “common neighb.” property Proof: Part - setup. 2 common neighborhood (in ) too small

36. Vertices which do not fulfill the “common neighb.” property Proof: Part - setup. 2 “Too many” missing edges:

37. Proof: Part - steps 2 • All „exceptional sets“ are small (next slide) • „Remainder“ is very sparse (in two slides) • Bootstrapping … (details omitted)

38. Example: is small

39. General case: • Inclusion/Exclusion • We obtain a • contradiction as • soon as Example: is small • Missing edges: • at least

40. Red graph is very sparse • Suppose that we know more: maxdeg • Idea: remove all red edges, add all missing edges • disjoint blue edges in !

41. …and is an optimal bipartition! We say that has gap 1. In general: red edges: has gap Remark: ! Proof: Part . 3 • Suppose • How does a maximum triangle-free subgraph look like? up to one edge!

42. Then Proof: Part - continued 3 • Fix any optimal partition • There are disjoint triangles with and ... • Problem: there might be exponentially many optimal bipartitions!

43. Problem - Erdös (1976) Conjecture (Erdös 1976): Everytriangle-free graph on n vertices can be made bipartite by deleting at most n2/25 edges. Note: If true, then best possible.