arXiv subjects

Rade Živaljević

Publications and source records attributed to Rade Živaljević.

13 recordsLinked to original sources

arXiv2026

On a class of toric manifolds arising from simplicial complexes

Given an arbitrary abstract simplicial complex $K$ on $[m]:=\{1,2,\ldots, m\}$, different from the simplex $Δ_{[m]}$ with $m$ vertices, we introduce and study a canonical $(2m-2)$-dimensional toric manifold $X_K$, associated to the canonical $(m-1)$-dimensional complete regular fan $Σ_K$. This construction yields an infinite family of toric manifolds that are not quasitoric and provides a topological proof of the Dehn-Sommerville relations for the associated Bier sphere $\mathrm{Bier}(K)$. Finally, we classify the canonical real and complex moment-angle manifolds of Lusternik-Schnirelmann category $\leq 2$ and prove a criterion for orientability of the canonical real toric manifolds.

math.AT↗

arXiv2025

Lower bounds on the number of envy-free divisions

We analyze lower bounds for the number of envy-free divisions, in the classical Woodall-Stormquist setting and in a non-classical case, when envy-freeness is combined with the equipartition of a measure. 1. In the first scenario, there are $r$ hungry players, and the cake (that is, the segment $[0,1]$) is cut into $r$ pieces. Then there exist at least two different envy-free divisions. This bound is sharp: for each $r$, we present an example of preferences such that there are exactly two envy-free divisions. 2. In the second (hybrid) scenario, there are $p$ not necessarily hungry players ($p$ is a prime) and a continuous measure $μ$ on $[0,1]$. The cake is cut into $2p-1$ pieces, the pieces are allocated to $p$ boxes (with some restrictions) and the players choose the boxes. Then there exists at least $\binom{2p-1}{p-1} \cdot 2^{2-p}$ envy-free divisions such that the measure $μ$ is equidistributed among the players.

math.CO↗

arXiv2024

Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture

We prove that a Murai sphere is flag if and only if it is a nerve complex of a flag nestohedron and classify all the polytopes arising in this way. Our classification implies that flag Murai spheres satisfy the Nevo-Petersen conjecture on $γ$-vectors of flag homology spheres. We continue by showing that a Bier sphere is minimally non-Golod if and only if it is a nerve complex of a truncation polytope different from a simplex and classify all the polytopes arising in this way. Finally, the notion of a cubical Bier sphere is introduced based on the polyhedral product construction, and we study combinatorial and geometrical properties of these cubical complexes.

math.CO↗

arXiv2024

Ky Fan theorem for sphere bundles

The classic Ky Fan theorem is a combinatorial equivalent of Borsuk-Ulam theorem. It is a generalization and extension of Tucker's lemma and, just like its predecessor, it pinpoints important properties of antipodal colorings of vertices of a triangulated sphere $S^n$. Here we describe generalizations of Ky Fan theorem for the case when the sphere is replaced by the total space of a triangulated sphere bundle.

math.CO↗

arXiv2023

Cooperative envy-free division

Relying on configuration spaces and equivariant topology, we study a general "cooperative envy-free division problem". A group of players want to cut a "cake" $I=[0,1]$ and divide among themselves the pieces in an envy-free manner. Once the cake is cut and served in plates on a round table (at most one piece per plate), each player makes her choice by pointing at one (or several) plates she prefers. The novelty is that her choice may depend on the whole "allocation configuration". In particular, a player may choose an empty plate (possibly preferring one of the empty plates over the other), and take into account not only the content of her preferred plate, but also the content of the neighbouring plates. We show that if the number of players is a prime power, in this setting an envy-free division exists under standard assumptions that the preferences are closed.

math.CO↗

arXiv2022

Envy-free division in the presence of a dragon

We prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive) player, called the "dragon". There are two basic scenarios. 1. There are $r-1$ players and a dragon. Once the "cake" is divided into $r$ parts, the dragon makes his choice and grabs one of the pieces. After that the players want to divide the remaining pieces in an envy-free fashion. 2. There are $r+1$ players who divide the cake into $r$ pieces. A ferocious dragon comes and swallows one of the players. The players want to cut the cake in advance in such a way that no matter who is the unlucky player swallowed by the dragon, the remaining players can share the tiles in an envy-free manner. In both settings the players are allowed to choose degenerate pieces of the cake. Moreover, they construct in advance both a cut of the cake and a "decision tree", allowing them to minimize the uncertainty of what pieces can be given to each of the players.

math.CO↗

arXiv2021

Envy-free division via configuration spaces

The classical approach to envy-free division and equilibrium problems relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology. We illustrate the method by proving several relatives (extensions) of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.

math.CO↗

arXiv2020

Splitting necklaces, with constraints

We prove several versions of N. 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 (including "degenerate pieces" if they exist), provided the number of thieves $r=p^ν$ is a prime power. (2) The "binary splitting theorem" claims that if $r=2^d$ 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 [7]) in the case $r=2^d$. (3) An interesting variation arises when the thieves have their own individual preferences. We prove several "envy-free fair necklace-splitting theorems" of various level of generality. By specialization we obtain numerous corollaries, among them envy-free versions of (a) "almost equicardinal splitting theorem", (b) "necklace-splitting theorem for $r$-unavoidable preferences", (c) "envy-free binary splitting theorem", etc. As a corollary we also obtain a recent result of Avvakumov and Karasev [1] about envy-free divisions where players may prefer an empty part of the necklace.

math.CO↗

arXiv2020

Topologically sensitive metaheuristics

This paper proposes topologically sensitive metaheuristics, and describes conceptual design of topologically sensitive Variable Neighborhood Search method (TVNS) and topologically sensitive Electromagnetism Metaheuristic (TEM).

cs.AI↗

arXiv2018

A Tverberg type theorem for collectively unavoidable complexes

We prove (Theorem 2.4) that the symmetrized deleted join $SymmDelJoin(\mathcal{K})$ of a "balanced family" $\mathcal{K} = \langle K_i\rangle_{i=1}^r$ of collectively $r$-unavoidable subcomplexes of $2^{[m]}$ is $(m-r-1)$-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result (Theorem 3.2) which is more conceptual and more general then previously known results. Already the case $r=2$ of Theorem 3.2 seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman's discrete Morse theory.

math.CO↗

arXiv2017

Alexander $r$-tuples and Bier complexes

Alexander $r$-tuples are introduced as a common generalization of pairs of Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes of Blagojević, Frick and Ziegler. The associated "Bier complexes" include both the Bier spheres and "optimal multiple chessboard complexes" as interesting, special cases. Our main result is 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 collectively unavoidable $r$-tuple is $(n-r-1)$-connected. We also give a complete classification (Theorem 5.1 and Corollary 5.2) of Alexander $r$-tuples and Bier complexes.

math.CO↗

arXiv2016

Symmetric multiple chessboard complexes and a new theorem of Tverberg type

We prove a new theorem of Tverberg type which confirms the conjecture of Blagojevic, Frick, and Ziegler about the existence of "balanced Tverberg partitions" (Conjecture 6.6 in, Tverberg plus constraints, Bull. London Math. Soc., 46 (2014) 953-967). The proof relies on the connectivity and shellability properties of multiple chessboard complexes and their symmetric analogues.

math.CO↗

arXiv2015

Colorful versions of the Lebesgue, KKM, and Hex theorem

Following and developing ideas of R. Karasev (Covering dimension using toric varieties, arXiv:1307.3437), we extend the Lebesgue theorem (on covers of cubes) and the Knaster-Kuratowski-Mazurkiewicz theorem (on covers of simplices) to different classes of convex polytopes (colored in the sense of M. Joswig). We also show that the $n$-dimensional Hex theorem admits a generalization where the $n$-dimensional cube is replaced by a $n$-colorable simple polytope. The use of quasitoric manifolds offers great flexibility and versatility in applying the general method.

math.MG↗