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Proof of the middle levels conjecture

Proof of the middle levels conjecture. Torsten Mütze. The middle layer graph. Consider the cube. 11...1. 111. level 3. level 2. 110. 101. 011. level 1. 100. 010. 001. level 0. 000. 00... 0. Middle layer graph. The middle layer graph. Middle layer of . 10110. 10101.

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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!

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