Proof of the middle levels conjecture

# Proof of the middle levels conjecture

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## Proof of the middle levels conjecture

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1. Proof of the middle levels conjecture Torsten Mütze

2. The middlelayergraph • Consider the cube 11...1 111 • level 3 • level 2 110 101 011 • level 1 100 010 001 • level 0 000 00...0 Middlelayergraph

3. The middlelayergraph • Middlelayer of 10110 10101 01101 01011 00111 11100 11010 11001 10011 01110 10001 01100 00110 00101 00011 11000 10100 10010 01010 01001 • bipartite, connected • number of vertices: • degree:

4. The middlelayergraph • Middlelayer of 10110 10101 01101 01011 00111 11100 11010 11001 10011 01110 10001 01100 00110 00101 00011 11000 10100 10010 01010 01001 • bipartite, connected • number of vertices: • degree: • automorphisms:bitpermutation + inversion,

5. The middlelayergraph • Middlelayer of 10110 10101 01101 01011 00111 11100 11010 11001 10011 01110 10001 01100 00110 00101 00011 11000 10100 10010 01010 01001 • bipartite, connected • number of vertices: • degree: • automorphisms:bitpermutation+ inversion,

6. The middlelayergraph • Middlelayer of 10110 10101 01101 01011 00111 11100 11010 11001 10011 01110 10001 01100 00110 00101 00011 11000 10100 10010 01010 01001 • bipartite, connected • number of vertices: • degree: • automorphisms:bitpermutation + inversion, • vertex-transitive

7. The middlelevelsconjecture Conjecture: The middlelayergraphcontains a Hamilton cycleforevery . • probablyfirstmentioned in [Havel 83], [Buck, Wiedemann 84] • also attributed to Dejter, Erdős, Trotter[Kierstead, Trotter 88]and variousothers • exercise (!!!) in [Knuth 05]

8. The middlelevelsconjecture Conjecture: The middlelayergraphcontains a Hamilton cycleforevery . • Motivation: • Gray codes • Conjecture[Lovász 70]: Everyconnectedvertex-transitivegraphcontains a Hamilton path.

9. History of the conjecture Numericalevidence: The conjectureholdsfor all [Moews, Reid 99], [Shields, Savage 99],[Shields, Shields, Savage 09], [Shimada, Amano 11]

10. History of the conjecture • Asymptoticresults: • The middlelayergraphcontains a cycle of length • [Savage 93] • [Felsner, Trotter 95] • [Shields, Winkler 95] • [Johnson 04]

11. History of the conjecture Otherrelaxations and partial results: [Kierstead, Trotter 88][Duffus, Sands, Woodrow 88][Dejter, Cordova, Quintana 88][Duffus, Kierstead, Snevily 94][Hurlbert 94][Horák, Kaiser, Rosenfeld, Ryjácek 05][Gregor, Škrekovski 10]…

12. Ourresults Theorem 1: The middlelayergraphcontains a Hamilton cycleforevery . Theorem 2: The middlelayergraphcontains different Hamilton cycles. Remarks: number of automorphismsisonly ,so Theorem 2 isnot an immediate consequence of Theorem 1

13. Ourresults Theorem 1: The middlelayergraphcontains a Hamilton cycleforevery . Theorem 2: The middlelayergraphcontains different Hamilton cycles. Remarks: number of Hamilton cyclesis at most ,so Theorem 2 is best possible

14. Proofideas Step 1:Build a 2-factor in the middlelayergraph Step 2: Connect the cycles in the 2-factor to a singlecycle

15. Structure of the middlelayergraph

16. Structure of the middlelayergraph

17. Structure of the middlelayergraph A Hamilton cycle Catalannumbers

18. Structure of the middlelayergraph A Hamilton cycle

19. Structure of the middlelayergraph A Hamilton cycle

20. Structure of the middlelayergraph A Hamilton cycle

21. Structure of the middlelayergraph A Hamilton cycle

22. Step 1: Build a 2-factor Constructionfrom[M., Weber 12] isomorphism (bitpermutation + inversion) ???

23. Step 1: Build a 2-factor Constructionfrom[M., Weber 12] 2-factor isomorphism (bitpermutation + inversion)

24. Step 1: Build a 2-factor Constructionfrom[M., Weber 12] • parametrizingyields different 2-factors • essentiallyonlyonecanbeanalyzed: = plane treeswithedges 2-factor Fundamental problem:varyingchangesglobally

25. Step 2: Connect thecycles New ingredient: Flippablepairs is a flippablepair,ifthereis a flipped pair 2-factor such that

26. Step 2: Connect thecycles New ingredient: Flippablepairs is a flippablepair,ifthereis a flipped pair 2-factor such that

27. Step 2: Connect thecycles New ingredient: Flippablepairs 2-factor flippable pairsyielddifferent 2-factors + verypreciselocalcontrol …wecanconstructmanyflippablepairs

28. Step 2: Connect thecycles New ingredient: Flippablepairs 2-factor Auxiliarygraph

29. Step 2: Connect thecycles Lemma 1:Ifisconnected, then the middlelayergraphcontains a Hamilton cycle. Lemma 2:Ifcontains different spanningtrees, then the middlelayergraphcontains different Hamilton cycles. 2-factor Auxiliarygraph

30. The crucialreduction Provethat isconnected (has manyspanningtrees) Provethat middlelayergraph contains a Hamilton cycle (many Hamilton cycles)

31. Analysis of 2 leaves 6 leaves 5 leaves 4 leaves 3 leaves = plane treeswithedges

32. Analysis of 2 leaves 6 leaves 5 leaves 4 leaves 3 leaves = plane treeswithedges

33. Thankyou!