1. Explicit matchings in the middle levels graph Torsten Mütze

2. The middle levels graph Why are we interested in finding perfect matchings in ? Conjecture (Erdös, Trotter, Dejter): For any n, the middle levels graph has a Hamiltonian cycle. The discrete cube Q n Vertex set: all bitstrings of length n Edge set: connect two vertices iff they differ in exactly one bit

3. What is known about the conjecture Computer experiments for small n : The conjecture holds for all [Shields, Shields, Savage ‘06] The number of vertices of is huge: General results: For any n, the graph has a cycle containing at least a … … 0.893-fraction of all vertices [Savage, Winkler ‘95] … (1-o(1))-fraction of all vertices [Johnson ‘04] One possible approach: Any Hamiltonian cycle in is the union of two perfect matchings Find large families of perfect matchings in and try to combine two of them This talk: review and compare several known constructions 001010100 011101011

4. is bipartite and ( n +1)-regular By Hall‘s theorem a perfect matching in exists, even a 1-factorization into n +1 disjoint perfect matchings Perfect matchings exist 101001001010001011000101001001001100010100011 10110101011001101110011010101100111 11000 111001101011001 n +1 n

5. is bipartite and ( n +1)-regular By Hall‘s theorem a perfect matching in exists, even a 1-factorization into n +1 disjoint perfect matchings Show me an ‚explicit‘ matching! 11000 111001101011001 101001001010001011000101001001001100010100011 10110101011001101110011010101100111 Perfect matchings exist n +1 n

6. Construction 1 An inductive construction due to [Greene, Kleitman ’76], [de Bruijn, Tengbergen, Kruyswijk ’51]: It yields something much stronger, a symmetric chain decomposition of Q n For odd n, the chains induce a perfect matching in the middle levels graph 11…1 00…0 QnQn 1001 11 00

7. 1001 11 00 Construction 1 An inductive construction due to [Greene, Kleitman ’76], [de Bruijn, Tengbergen, Kruyswijk ’51]: It yields something much stronger, a symmetric chain decomposition of Q n For odd n, the chains induce a perfect matching in the middle levels graph 11…1 00…0 QnQn

8. Disadvantage: need to know previous construction steps Construction 1 An inductive construction due to [Greene, Kleitman ’76], [de Bruijn, Tengbergen, Kruyswijk ’51]: It yields something much stronger, a symmetric chain decomposition of Q n For odd n, the chains induce a perfect matching in the middle levels graph 11…1 00…0 QnQn 1001 11 00

9. Construction 2 A greedy construction due to [Aigner ’73]: In each layer sort vertices lexicographically: x < y if first at first position where they differ x i =1 and y i =0 Go through vertices in and greedily match them to first unmatched vertex in (if possible) Need to prove that this always yields a maximum matching (i.e. a perfect matching in the middle levels graph) Disadvantage: need to know previously matched vertices 1100 1010 1001 0110 0101 0011 1000 0100 0010 0001 1110 1101 1011 0111 1111 0000 QnQn

10. Construction 3 An explicit construction due to [Aigner ’73], [Greene, Kleitman ’76]: Repeatedly match closest 0 - and 1 -digits Flip first unmatched 0  1 (if there are any) 11001011100100 11100011001011110100

11. Observation: Bit-flipping does not change matched 0 - 1 -pairs Reverse operation: Flip last unmatched 1  0 (if there are any), in particular we indeed obtain a matching This matching is also maximum (perfect matching in ) We in fact obtain a symmetric chain decomposition Construction 3 11001011100100 11100011001011110100 11001011110110 11001011110111 11001011000100 10001011000100 00001011000100

12. 1100 1010 1001 0110 0101 0011 1000 0100 0010 0001 1110 1101 1011 0111 111100 00 QnQn Observation: Bit-flipping does not change matched 0 - 1 -pairs Reverse operation: Flip last unmatched 1  0 (if there are any), in particular we indeed obtain a matching This matching is also maximum (perfect matching in ) We in fact obtain a symmetric chain decomposition Construction 3

13. Equivalence of the constructions Theorem: All 3 constructions yield the same matchings (and SCDs): Depending on the context, each of the descriptions has its merits Construction 1: inductive SCD Construction 2: greedy matching Construction 3: explicit 0 - 1 -pairs approach

14. How to obtain more matchings So far we have found only one explicit matching in the middle levels graph How to obtain more? Idea: Apply automorphisms of : bit permutations + bit inversion So how many different matchings do we get? many different matchings, called lexicographic matchings Cyclic bit shifts yield the same matching Bit inversion does not produce any new matchings

15. Do they yield a Hamiltonian cycle? Theorem ([Duffus, Sands, Woodrow ’88]): For, no two lexicographic matchings form a Hamiltonian cycle in. Not true for the trivial case (2 lexicographic matchings): Larger families of explicit matchings? 001010100 011101011

16. Lexical Matchings Superset of the lexicographic matchings 110010100 110010100 110010100 110010100 110010100 110010100 0 3 2 1 4 u ( x ) := number of upsteps below the x-axis [Kierstead, Trotter ‘88] i -lexical matching := for each flip the bit for which u ( x )= i The i -lexical matchings, i =0,1,…, n, form a 1-factorization of

17. Superset of the lexicographic matchings 110010100 110010100 Lexical Matchings 111010100 i -lexical matching := for each flip the bit for which u ( x )= i u ( x ) := number of upsteps below the x-axis 010010110 101010100 110010100 111010000 Other potential partners for y : 2 4 3 1 0 by the same argument as before So we indeed get a matching [Kierstead, Trotter ‘88]

18. Superset of the lexicographic matchings: 0 -lexical matching = lexicographic matching Behavior under automorphisms of the middle levels graph: Lexical Matchings different matchings (the lexicographic matchings) another different matchings 0 -lexical matching 1 -lexical matching n /2 -lexical matching n -lexical matching another different matchings No new matchings [Kierstead, Trotter ‘88] many different matchings, called lexical matchings Do two of them form a Hamiltonian cycle? n =1,2,3: yes, n =4: no, n >4: not known

19. Outlook Modular matchings [Duffus, Kierstead, Snevily ‘94] defined via an algebraic construction Similarly large class like the lexical matchings, distinct from the lexical matchings Similar behavior under automorphisms of the middle levels graph Do two of them form a Hamiltonian cycle? Not known.

20. Thank you! Questions?