Citation key |
DaeubelJaegerMuetzeScheucher2019 |

Author |
Däubel, Karl and Jäger, Sven and Mütze, Torsten and Scheucher, Manfred |

Title of Book |
Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB) |

Pages |
611–618 |

Year |
2019 |

ISSN |
0862-9544 |

Location |
Bratislava |

Journal |
Acta Mathematica Universitatis Comenianae |

Volume |
88 |

Number |
3 |

Month |
August |

Note |
extended abstract |

Editor |
Nešetřil, Jaroslav and Škoviera, Martin |

Abstract |
The n-cube is the poset obtained by ordering all subsets of 1,...,n by inclusion, and it can be partitioned into n choose ⌊n/2⌋ chains, which is the minimum possible number. Two such decompositions of the n-cube are called orthogonal if any two chains of the decomposition share at most a single element. Shearer and Kleitman conjectured in 1979 that the n-cube has ⌊n/2⌋+1 pairwise orthogonal decompositions into the minimum possible number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the n-cube has three pairwise orthogonal chain decompositions for n ≥ 24. In this paper, we construct four pairwise orthogonal chain decompositions of the n-cube for n ≥ 60. We also construct five pairwise edge-disjoint symmetric chain decompositions of the n-cube for n ≥ 90, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, Jäger, Mütze, Sawada, and Wille. |