arXiv subjects

Siniša T. Vrećica

Publications and source records attributed to Siniša T. Vrećica.

6 recordsLinked to original sources

arXiv2020

Generalized chessboard complexes and discrete Morse theory

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 the Bier sphere of an associated simplicial complex $K$. We illustrate the use of "standard discrete Morse functions" on generalized chessboard complexes by proving a connectivity result for chessboard complexes with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for $j$-wise disjoint partitions of a simplex.

math.MG↗

arXiv2019

Combinatorics of `unavoidable complexes'

The partition number $π(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojević, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.

math.CO↗

arXiv2018

Topology of unavoidable complexes

The partition number $π(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if for each continuous map $f: | K| \rightarrow \mathbb{R}^d$ there exist $r$ vertex disjoint faces $σ_1,\ldots, σ_r$ of $| K|$ such that $f(σ_1)\cap\ldots\cap f(σ_r)\neq\emptyset$. Motivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.6, 3.9, 4.6) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (Theorem 4.6), summarizing and extending results of G. Schild, B. Grünbaum and many others, is that interesting examples of (globally) $r$-non-embeddable complexes can be found among the joins $K = K_1\ast\ldots\ast K_s$ of $r$-unavoidable complexes.

math.AT↗

arXiv2016

Hyperplane mass equipartition problem and the shielding functions of Ramos

We give a proof of the result of Edgar Ramos which claims that two finite, continuous Borel measures $μ_1$ and $μ_2$ defined on $\mathbb{R}^5$ admit an equipartition by a collection of three hyperplanes. Our proof illuminates one of the central methods developed and used in our earlier papers and may serve as a good `test case' for addressing (and resolving) the `issues' raised in the paper "Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem", arXiv:1502.02975 [math.AT]. We also offer a degree-theoretic interpretation of the `parity calculation method' developed by Ramos and demonstrate that, up to minor corrections or modifications, it remains a rigorous and powerful tool for proving results about mass equipartitions.

math.MG↗

arXiv2015

Multiple chessboard complexes and the colored Tverberg problem

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 applications in Tverberg type problems where multiple points of the same color are permitted. These results also provide a foundation for the new results of Tverberg-van Kampen-Flores type, as announced in arXiv:1502.05290 [math.CO].

math.CO↗

arXiv2007

Local chromatic number and distinguishing the strength of topological obstructions

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the homomorphism complex Hom(K_2,G) and its suspension, respectively. These investigations follow the line of research initiated by Matousek and Ziegler who recognized a hierarchy of the different topological expressions that can serve as lower bounds for the chromatic number of a graph. Our results imply that the local chromatic number of 4-chromatic Kneser, Schrijver, Borsuk, and generalized Mycielski graphs is 4, and more generally, that 2r-chromatic versions of these graphs have local chromatic number at least r+2. This lower bound is tight in several cases by results in an earlier paper of the first two authors.

math.CO↗