Dusko Jojic

University of Banja Luka

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...

We prove a multiple coloured Tverberg theorem and a balanced coloured Tverberg theorem , applying different methods, tools and ideas. The proof of the first theorem uses a multiple chessboard complex (as configuration space) and the Eilenberg–Krasnoselskii theory of degrees of equivariant maps for non-free group actions. The proof of the second res...

Мы доказываем кратную цветную теорему Тверберга и сбалансированную цветную теорему Тверберга, пользуясь различными методами и приемами. Доказательство первой теоремы использует в качестве конфигурационнго пространства шахматный комплекс с кратностями и теорию Эйленберга-Красносельского о степенях эквивариантных отображений для несвободных действий...

We prove that the symmetrized deleted join SymmDelJoin(\(\mathcal{K}\) ) of a “balanced family” \(\mathcal{K}\) = 〈Ki〉 ri=1 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 which is more conceptual and more general than previously known results. Already the...

The type A colored Tverberg theorem of Blagojevi\'{c}, Matschke, and Ziegler provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices is a prime number. We extend this result to an optimal, type A colored Tverberg theorem for multisets of colored points, which is valid for each...

The partition invariant \(\pi (K)\) of a simplicial complex \(K\subseteq 2^{[m]}\) is the minimum integer \(\nu \), such that for each partition \(A_1\uplus \cdots \uplus A_\nu = [m]\) of [m], at least one of the sets \(A_i\) is in K. A complex K is r-unavoidable if \(\pi (K)\le r\). We say that a complex K is almost r-non-embeddable in \({\mathbb...

Шахматные комплексы и их обобщения, как объекты, и дискретная теория Морса, как инструмент, представлены в виде объединяющей темы, связывающая различные области геометрии, топологии, алгебры и комбинаторики. Теорема Эдмондса и Фулкерсона о бутылочном горлышке (минимаксе) реализуется и интерпретируется как результат о критической точке дискретной фу...

Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted as a result about a critical point of a discrete Morse function on t...

We prove a "multiple colored Tverberg theorem" and a "balanced colored Tverberg theorem", by applying different methods, tools and ideas. The proof of the first theorem uses multiple chessboard complexes (as configuration spaces) and Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions. The proof of the second result r...

We prove several versions of Alon's "necklace-splitting theorem", subject to additional constraints. For illustration the "Equicardinal necklace-splitting theorem" claims that, without increasing the number of cuts, one can guarantee that each thief is allocated (approximately) the same number of pieces of the necklace. Our main topological tool ar...

Ehrenborg noted that all tilings of a bipartite planar graph are encoded by its cubical matching complex and claimed that this complex is collapsible. We point out to an oversight in his proof and explain why these complexes can be the union of collapsible complexes. Also, we prove that all links in these complexes are suspensions up to homotopy. F...

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 ge...

We prove a new theorem of Tverberg–van Kampen–Flores type, which confirms a conjecture of Blagojević et al. about the existence of ‘balanced Tverberg partitions’ (Conjecture 6.6 in [Tverberg plus constraints, Bull. London Math. Soc. 46:953–967 (2014]). The conditions in this theorem are somewhat weaker than in the original conjecture, and we show t...

The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $\pi(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by th...

Alexander $r$-tuples are introduced as a common generalization of pairs of Alexander dual complexes (Alexander $2$-tuples) and $r$-unavoidable complexes of Blagojevi\'{c}, 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 The...

The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf algebra of hypergraphs. We characterize a wide class of eulerian hypergraphs according to the combinatorics of unde...

For a given tree T we consider the facial structure of the acyclic Birkhoff polytope . We also determine the f-vector of the polytope consisting of all tridiagonal doubly stochastic matrices of order n. Finally, we count the number of combinatorially distinct faces of in each dimension.

The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $\pi(K)\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if fo...

Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of $GL$-Complexes, Journal of Algebra 2001) we study the connectivity degree and shellability of multiple chessboard complexes. Our central new results (Theorems 3.2 and 4.4) provide sharp connectivity bounds relevant to appl...

The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf algebra of hypergraphs. The characterizations of eulerian hypergraphs are obtained according to the combinatorics o...

The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the cross...

For every directed graph D we consider the complex of all directed subforests δ(D). The investigation of these complexes was started by D. Kozlov. We generalize a result of Kozlov and prove that complexes of directed trees of complete multipartite graphs are shellable. We determine the h-vector of δ(-K→m,n) and thehomotopy type of δ(K→n1,n2,...,nk)...

The question of shellability of complexes of directed trees was asked by R. Stanley. D. Kozlov showed that the existence of a complete source in a directed graph provides a shelling of its complex of directed trees. We will show that this property gives a shelling that is straightforward in some sense. Among the simplicial polytopes, only the cross...

Given a graded poset P, let I(P) denote the associated poset of intervals and Et(P) the poset obtained from P by the Et-construction of Paffenholz and Ziegler [7]. We analyze how the ab-index behaves under those operations and prove that its change is expressed in terms of certain, quite explicit, recursively defined linear operators. If the poset...

Weighted derivations W 1 and W 2 allowed R. Ehrenborg and M. Readdy (Discrete Comput. Geom. 21:389–403, [1999]) to give a recursive description of the cd-indices of the lattices of the regions of the arrangements $\mathcal{A}_{n}$ and ℬn . In part motivated by this, we describe a new basis for the subspace spanned by ab-indices of all simplicial...

This paper is within the Bishop's constructive mathematics. A quasi-antiorder relational system means a pair (A, R) where (A,=, �=) is a set with apartness and R is a consistent and cotransitive binary relation on A. We define and study a quotient system mapping ϕ such that the factor relation R/Cokerϕ on the factor set A/Cokerϕ is also a quasi-ant...

We show that the stellar subdivisions of a simplex are extendably shellable. These polytopes appear as the facets of the dual of a hypersim- plex. Using this fact, we calculate the simplicial and toric h-vector of the dual of a hypersimplex. Finally, we calculate the contribution of each shelling component to the toric h-vector.

We show that the stellar subdivisions of a simplex are extendably shellable. These polytopes appear as the facets of the dual of a hypersimplex. Using this fact, we calculate the simplicial and toric h-vector of the dual of a hypersimplex. Finally, we calculate the contribution of each shelling component to the toric h-vector.

We will show that shellability, Cohen-Macaulayness and vertex-de composability of a graded, planar poset P are all equivalent with the fact that P has the maximal possible number of edges. Also, for a such poset we will find an R−labelling with {1, 2} as the set of labels. Using this, we will obtain all essential linear inequalities for the flag h−...