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Marek Filakovský

Publications and source records attributed to Marek Filakovský.

8 recordsLinked to original sources
arXiv2026

The Topology of $k$-Robust Clique Complexes in Grid-like Graphs

We study $k$-robust clique complexes, a family of simplicial complexes that generalizes the traditional clique complex. Here, a subset of vertices forms a simplex provided it does not contain an independent set of size $k$. We investigate these complexes for square sequence graphs, a class of bipartite graphs introduced here that are constructed by iteratively attaching $C_4$ cycles. This class includes rectangular grid graphs $G_{m,n}$. We show that for $k=2$ and $k=3$, the homotopy type is a wedge sum of $(2k-3)$-dimensional spheres, a result we extend to arbitrary $k$ under specific structural constraints on the attachment sequence. Our approach utilizes König's theorem to decompose the complex into manageable components, whose homotopy types are easy to understand. This then enables an inductive proof based on the decomposition and standard tools of algebraic topology. Finally, we utilize Alexander duality to connect our results to the study of total-$k$-cut complexes, generalizing recent results concerning the homotopy types of total-$k$-cut complexes for grid graphs.

math.CO↗
arXiv2026

Curves on the torus with few intersections

Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$. We determine the maximum size of such a set for every k. In particular, the maximum never exceeds k+6, and it does not exceed k+4 when k is large. As this quantity coincides with the maximal number of columns of a generic k-modular matrix with two rows, our result also settles the column number problem, a problem of interest in combinatorial optimization, for such matrices.

math.CO↗
arXiv2025

Hardness of 4-Colourings G-Colourable Graphs

We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite loop-less graph H. Brakensiek and Guruswami [SODA 2018] conjectured the hardness extends to promise graph homomorphism problems as follows: fix a pair of non-bipartite loop-less graphs G, H such that there is a homomorphism from G to H, it is NP-hard to distinguish between graphs that are G-colourable and those that are not H-colourable. We confirm this conjecture in the cases when both G and H are 4-colourable. This is a common generalisation of previous results of Khanna, Linial, and Safra [Comb. 20(3): 393-415 (2000)] and of Krokhin and Opršal [FOCS 2019]. The result is obtained by combining the algebraic approach to promise constraint satisfaction with methods of topological combinatorics and equivariant obstruction theory.

cs.CC↗
arXiv2023

Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023).

cs.CC↗
arXiv2022

Computing homotopy classes for diagrams

We present an algorithm that, given finite simplicial sets $X$, $A$, $Y$ with an action of a finite group $G$, computes the set $[X,Y]^A_G$ of homotopy classes of equivariant maps $\ell \colon X \to Y$ extending a given equivariant map $f \colon A \to Y$ under the stability assumption $\dim X^H \leq 2 \operatorname{conn} Y^H$ and $\operatorname{conn} Y^H \geq 1$, for all subgroups $H\leq G$. For fixed $n = \operatorname{dim} X$, the algorithm runs in polynomial time. When the stability condition is dropped, the problem is undecidable already in the non-equivariant setting. The algorithm is obtained as a special case of a more general result: For finite diagrams of simplicial sets $X$, $A$, $Y$, i.e. functors $\mathcal{I}^\mathrm{op} \to \mathsf{sSet}$, in the stable range $\operatorname{dim} X \leq 2 \operatorname{conn} Y$ and $\operatorname{conn} Y > 1$, we give an algorithm that computes the set $[X, Y]^A$ of homotopy classes of maps of diagrams $\ell \colon X \to Y$ extending a given $f \colon A \to Y$. Again, for fixed $n = \dim X$, the running time of the algorithm is polynomial. The algorithm can be utilized to compute homotopy invariants in the equivariant setting -- for example, one can algorithmically compute equivariant stable homotopy groups. Further, one can apply the result to solve problems from computational topology, which we showcase on the following Tverberg-type problem: Given a $k$-dimensional simplicial complex $K$, is there a map $K \to \mathbb{R}^{d}$ without $r$-tuple intersection points? In the metastable range of dimensions, $rd \geq (r+1)k +3$, the result of Mabillard and Wagner shows this problem equivalent to the existence of a particular equivariant map. In this range, our algorithm is applicable and, thus, the $r$-Tverberg problem is algorithmically decidable (in polynomial time when $k$, $d$ and $r$ are fixed).

math.AT↗
arXiv2014

Effective homology for homotopy colimit and cofibrant replacement

We extend the notion of simplicial set with effective homology to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal{I} \to \mathsf{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mathsf{hocolim} X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^\mathsf{cof}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.

math.AT↗
arXiv2013

Are two given maps homotopic? An algorithmic viewpoint

This paper presents two algorithms. In their simplest form, the first algorithm decides the existence of a pointed homotopy between given simplicial maps f, g from X to Y and the second computes the group $[ΣX,Y]^*$ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected and the algorithms run in polynomial time when the dimension of X is fixed. More generally, these algorithms work relative to a subspace A of X, fibrewise over a simply connected B and also equivariantly when all spaces are equipped with a free action of a fixed finite group G.

math.AT↗