Browsing by Author "Zivaljevic, Rade"

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    Alexander r-tuples and Bier complexes
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2016) Jojic, Dusko; Nekrasov, Ilya; Panina, Gaiane; Zivaljevic, Rade
    We introduce and study Alexander r-tuples K = Kiir i=1 of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins K delta of Alexander r-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the r-fold deleted join of Alexander r-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the r-fold deleted join of a collective unavoidable r-tuple is (n - r - 1)-connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander r-tuples and Bier complexes.
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    Splitting Necklaces, with Constraints
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2020) Jojic, Dusko; Panina, Gaiane; Zivaljevic, Rade
    We prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints, as illustrated by the following results. (1) The "almost equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one guarantees the existence of a fair splitting such that each thief is allocated (approximately) one and the same number of pieces of the necklace, provided the number of thieves r=pν is a prime power. (2) The "binary splitting theorem" claims that if r=2d and the thieves are associated with the vertices of a d-cube then, without increasing the number of cuts, one can guarantee the existence of a fair splitting such that adjacent pieces are allocated to thieves that share an edge of the cube. This result provides a positive answer to the "binary splitting necklace conjecture" of Asada at al. (Conjecture 2.11 in [5]) in the case r=2d.