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Djordje Baralic

Publications and source records attributed to Djordje Baralic.

At least 19 recordsLinked to original sources
arXiv2026

Polyominoes with maximal number of deep holes

In this paper, we study the extremal behaviour of deep holes in polyominoes. We determine the maximum number, $h_n$ of deep holes that an $n$-omino can enclose, ensuring that the boundary of each hole is disjoint from the boundaries of any other hole and from the outer boundary of the $n$-tile. Using the versatile application of Pick's theorem, we establish the lower and the upper bound for $h_n$, and show that $h_n=\frac{n}{3}+o(n)$ asymptotically. To further develop these results, we compute $h_n$ as a function of $n$ for an infinite subset of positive integers.

math.CO↗
arXiv2025

Magical Property of Fullerenes

Fullerenes are an allotrope of carbon having hollow, cage-like structure. Atoms in the molecule are arranged in pentagonal and hexagonal rings, such that each atom is connected to three other atoms. Simple polyhedra having only pentagonal and hexagonal faces are a mathematical model for fullerenes. We say that a fullerene with $n$ vertices has magical property if the numbers $1, 2, \dots, n$ may be assigned to its vertices so that the sums of the numbers in each pentagonal faces are equal and the sums of the numbers in each hexagonal faces are equal. We show that $C_{8n+4}$ does not admit such an arrangement for all $n$, while there are fullerenes, like $C_{24}$ and $C_{26}$ that have many nonisomorphic such arrangements.

math.CO↗
arXiv2024

Chromatic numbers for facet colouring of some generalised associahedra

The chromatic number related to a colouring of facets of certain classes of generalised associahedra is studied. The exact values are obtained for permutohedra, associahedra and simple permutoassociahedra, while lower and upper bounds are established for cyclohedra and stellohedra. The asymptotic values of the chromatic numbers for associahedra, cyclohedra and simple permutoassociahedra are given.

math.CO↗
arXiv2023

Mod $p$ Buchstaber invariant

We investigate the mod $p$ Buchstaber invariant of the skeleta of simplices, for a prime number $p$, and compare them for different values of $p$. For $p=2$, the invariant is the real Buchstaber invariant. Our findings reveal that these values are generally distinct. Additionally, we determine or estimate the mod $p$ Buchstaber invariants of certain universal simplicial complexes $X(\F_p^n)$.

math.AT↗
arXiv2023

Orthogonality related to Steiner and Pappus chains

The focus of this paper is on the study of specific circle formations known as orthogonal Pappus chains and the related incidence results that involve points of tangency between the circles in the construction. These chains give rise to new circle families, with their centres on a conic. The paper also explores similar questions for Steiner chains, although it is demonstrated that orthogonality cannot be defined for them. Nonetheless, explicit examples are provided, exhibiting comparable incidence results and raising whether two Steiner chains that share a common circle in the chains possess this property.

math.MG↗
arXiv2023

The Magic Permutohedron

We present a configuration called a magic permutohedron that shows the placement of the numbers of $\{1, 2, 3, \dots, 24\}$ in the vertices of the permutohedra so that the sum of numbers on each square side is 50 and the sum of the numbers in each hexagonal side is 75.

math.HO↗
arXiv2023

Existence of a small cover over a 15-colorable simple 4-polytope

The chromatic number for properly colouring the facets of a combinatorial simple $n$-polytope $P^n$ that is the orbit space of a quasitoric manifold satisfies the inequality $n\leq P^n\leq 2^n-1$. The inequality is sharp for $n=2$ but not for $n=3$ due to the Four Color theorem. In this note, we construct a simple 4-polytope admitting a characteristic map whose chromatic number equals $15$ and deduce that the predicted upper bound is attained for $n=4$. Analogues results are verified for the case of oriented small covers in dimensions $4$ and $5$.

math.CO↗
arXiv2021

Homology of polyomino tilings on flat surfaces

The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, for finite square grids on topological surfaces, the method of homology groups, namely the non-triviality of some specific element in the group allows a `coloring proof' of impossibility of a tiling. Several results about the non-existence of polyomino tilings on certain square-tiled surfaces are proved in the paper.

math.CO↗
arXiv2020

Universal simplicial complexes inspired by toric topology

Let $\mathbf{k}$ be the field $\mathbb{F}_p$ or the ring $\mathbb{Z}$. We study combinatorial and topological properties of the universal simplicial complexes $X(\mathbf{k}^n)$ and $K(\mathbf{k}^n)$ whose simplices are certain unimodular subsets of $\mathbf{k}^n$. As a main result we show that $X(\mathbf{k}^n)$, $K(\mathbf{k}^n)$ and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that $K(\mathbb{Z}^n)$ and $K(\mathbb{F}_2^n)$ are $(n-2)$-connected simplicial complexes. We discuss applications of these universal simplicial complexes to toric topology and number theory.

math.CO↗
arXiv2020

Proofs and surfaces

A formal sequent system dealing with Menelaus' configurations is introduced in this paper. The axiomatic sequents of the system stem from 2-cycles of Delta-complexes. The Euclidean and projective interpretations of the sequents are defined and a soundness result is proved. This system is decidable and its provable sequents deliver incidence results. A cyclic operad structure tied to this system is presented by generators and relations.

math.LO↗
arXiv2018

Spheres as Frobenius objects

Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension $d=1$, all the spheres are commutative Frobenius objects in categories whose arrows are ${(d+1)}$-dimensional cobordisms. With respect to the language of Frobenius objects, there is no distinction between these spheres---they are all free of additional equations formulated in this language. The corresponding structure makes out of the 0-dimensional sphere not a commutative but a symmetric Frobenius object. This sphere is mapped to a matrix Frobenius algebra by a 1-dimensional topological quantum field theory, which corresponds to the representation of a class of diagrammatic algebras given by Richard Brauer.

math.CT↗
arXiv2018

A simple permutoassociahedron

In the early 1990s, a family of combinatorial CW-complexes named permutoassociahedra was introduced by Kapranov, and it was realized by Reiner and Ziegler as a family of convex polytopes. The polytopes in this family are "hybrids" of permutohedra and associahedra. Since permutohedra and associahedra are simple, it is natural to search for a family of simple permutoassociahedra, which is still adequate for a topological proof of Mac Lane's coherence. This paper presents such a family.

math.CO↗
arXiv2017

Surprising Examples of Manifolds in Toric Topology!

We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The problem of cohomological rigidity for small covers is also studied and we have found a whole new series of weakly cohomologically rigid simple polytopes. New examples of manifolds provide the first known examples of quasitoric manifolds in higher dimensions whose orbit polytopes have chromatic numbers $χ(P^n)\geq 3n-5$.

math.AT↗
arXiv2014

A note on the combinatorial structure of finite and locally finite simplicial complexes of nonpositive curvature

We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of the complex onto a ball around this convex subcomplex. These projections will induce a convenient gradient matching on the complex. Besides we analyze the combinatorial structure of both CAT(0) and systolic locally finite simplicial complexes of arbitrary dimensions. We will show that both such complexes possess an arborescent structure. Along the way we make use of certain well known results regarding systolic geometry.

math.CO↗
arXiv2013

Around the Carnot theorem

We study the Carnot theorem and the configuration of points and lines in connection with it. It is proven that certain significant points in the configuration lie on the same lines and same conics. The proof of an equivalent statement formulated by Bradley is given. An open conjecture, established by Bradley, is proved using the theorems of Carnot and Menelaus.

math.AG↗
arXiv2013

Note on Totally Skew Embeddings of Quasitoric Manifolds over Cube

Totally skew embeddings are introduced by Ghomi and Tabachnikov. They are naturally related to classical problems in topology, such as the generalized vector field problem and the immersion problem for real projective spaces. In recent paper Topological obstructions to totally skew embeddings, {totaly skew} embeddings are studied by using the Stiefel-Whitney classes In the same paper it is conjectured that for every $n$-dimensional, compact smooth manifold $M^n$ $(n>1)$, $$N(M^n)\leq 4n-2α(n)+1,$$ where $N(M^n)$ is defined as the smallest dimension $N$ such that there exists a {\em totally skew} embedding of a smooth manifold $M^n$ in $\mathbb{R}^N$. We prove that for every $n$, there is a quasitoric manifold $Q^{2n}$ for which the orbit space of $T^n$ action is a cube $I^n$ and $$N(Q^{2n})\geq 8n-4α(n)+1.$$ Using the combinatorial properties of cohomology ring $H^* (Q^{2n}, \mathbb{Z}_2)$, we construct an interesting general non-trivial example different from known example of the product of complex projective spaces.

math.AT↗
arXiv2013

Cinderella, quadrilaterals and conics

We study quadrilaterals inscribed and circumscribed about conics. Our research is guided by experiments in software Cinderella. We extend the known results in projective geometry of conics and show how modern mathematical software brings new ideas in pure and applied mathematics. Poncelet theorem for quadrilaterals is proved by elementary means together with Poncelet's grid property.

math.AG↗