On symmetric chains and Hamilton cycles

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# On symmetric chains and Hamilton cycles

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## On symmetric chains and Hamilton cycles

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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!