On symmetric chains and Hamilton cycles

1. On symmetric chains and Hamilton cycles Torsten Mütze (based on jointworkwith Karl Däubel, Sven Jäger, Petr Gregor, Joe Sawada, Manfred Scheucher, and Kaja Wille)

2. The Booleanlattice • consider all subsets of orderedbyinclusion • a fundamental and widelystudiedposet • called-cube • -thlevel := all subsets of cardinality • itssizeis • 4-cube

3. The Booleanlattice • consider all subsets of orderedbyinclusion • a fundamental and widelystudiedposet • called-cube • -thlevel := all subsets of cardinality • itssizeis • odd • even • middlelevel(s)

4. Chain decompositions Theorem [Sperner 28]: The width (=size of a maximumantichain) of the -cubeisgivenby the size of itsmiddlelevel(s) . Theorem [Dilworth 50]: Anyposetcanbedecomposedintomanychains. • 4-cube • chaindecomposition

5. Symmetricchaindecompositions • usefulforapplications: symmetricchains, i.e., if a chainstarts at level , thenitends at level . • knownconstructions of SCDsfor the -cubedue to[De Bruijn, van Ebbenhorst Tengbergen, Kruiswijk 51], [Lewin 72], [Aigner 73], [White and Williamson 77], [Greene, Kleitman 76] • all constructionsyieldthe same SCD • notsymmetric • 4-cube • symmetric • chaindecomposition • (SCD)

6. Parenthesismatching • usefulforapplications: symmetricchains, i.e., if a chainstarts at level , thenitends at level . • knownconstructions of SCDsfor the -cubedue to[De Bruijn, van Ebbenhorst Tengbergen, Kruiswijk 51], [Lewin 72], [Aigner 73], [White and Williamson 77], [Greene, Kleitman 76] • all constructionsyieldthe same SCD • ‚parenthesismatching‘descriptionby[Greene, Kleitman 76] 1001110100 1001110110 1001110111 0001100100 1001100100

7. Edge-disjoint and orthogonal SCDs • Question: Are thereotherconstructions? • Definition:TwoSCDsareedge-disjoint, ifthey do not shareanyedges • Definition:TwoSCDsareorthogonal, ifanytwochainsintersect in at mostoneelement, except the twolongestchainsthatmayonlyintersect in and • 4-cube • 4-cube • Observe: orthogonal edge-disjoint

8. Edge-disjoint and orthogonal SCDs • Question:Howmanypairwiseedge-disjoint/orthogonal SCDscanwehopefor? • is anupperbound: • every SCD usesexactlyone of thoseedges • even • Theorem[Shearer, Kleitman 79]: The standardconstruction and itscomplementsaretwoorthogonal SCDs. • Conjecture[Shearer, Kleitman 79]: The -cube has pairwise orthogonal SCDs. • Theorem[Spink 17]: The -cube has threepairwise orthogonal SCDsfor .

9. Ourresults Theorem 1: The -cube has fourpw. orthogonal SCDsfor . Theorem 2: The -cube has fivepw. edge-disjointSCDsfor . Proof of Theorem 2: • Product lemma:If the -cube and -cubehaveedge-disjointSCDseach, then the -cube has edge-disjointSCDs. • find fiveedge-disjointSCDsfordimensions and • Fact: If and arecoprime, theneveryis a non-negative integer multiple of and . computer search in the necklace poset Proof of Theorem 1: • similar, but more complicated product lemma due to [Spink 17] • find four orthogonal SCDs for dimensions and

10. The centrallevelsproblem Middlelevelsconjecture: The subgraph of the-cubeinducedby the middletwolevels and has a Hamilton cycle. • -cube • problemwith a longhistory • answeredpositively in [M. 16] • Central levelsconjecture:The subgraph of the -cubeinducedbythe middlelevels has a Hamilton cycleforany . • raisedby[Savage93], [Gregor, Škrekovski 10], [Shen, Williams 15]

11. The centrallevelsproblem • -cube • Central levelsconjecture:The subgraph of the -cubeinducedbythe middlelevels has a Hamilton cycleforany . • known results: [Gray 53] [El-Hashash, Hassan 01], [Locke, Stong 03] • ??? [Gregor, Škrekovski 10] [M. 16]

12. Ourresults Theorem 3: The -cube has a Hamilton cyclethrough the middlefourlevels ( ) for all . Theorem 4: The -cube has a cyclefactorthrough the middlelevels for all and . • spanningcollectionof disjointcycles

13. The centrallevelsproblem Theorem 4: The -cube has a cyclefactorthrough the middlelevels for all and . Proof: • considertwoedge-disjointSCDs • as the dimensionisodd, all chainshaveoddlength, evenafterrestricting to middlelevels • takingevery second edge yieldstwoedge-disjointperfectmatchings • theirunionis a cyclefactor

14. Open problems known: four known: five • centrallevelsproblem: Can the cycles in the factorbejoined to a single Hamilton cycle?Structure of the cyclefactor? first open case: middle six levels • efficientalgorithms to generatethosecycles • exploitnew SCD constructions in otherapplications (Venndiagrams etc.) • Conjecture[Shearer, Kleitman 79]: The -cube has pairwise orthogonal SCDs. • weconjecturethat the -cube has pairwiseedge-disjointSCDs.

15. Thankyou!