arXiv subjects

Jan Kynčl

Publications and source records attributed to Jan Kynčl.

At least 19 recordsLinked to original sources

arXiv2025

Extending simple monotone drawings

We prove the following variant of Levi's Enlargement Lemma: for an arbitrary arrangement $\mathcal{A}$ of $x$-monotone pseudosegments in the plane and a pair of points $a,b$ with distinct $x$-coordinates and not on the same pseudosegment, there exists a simple $x$-monotone curve with endpoints $a,b$ that intersects every curve of $\mathcal{A}$ at most once. As a consequence, every simple monotone drawing of a graph can be extended to a simple monotone drawing of a complete graph. We also show that extending an arrangement of cylindrically monotone pseudosegments is not always possible; in fact, the corresponding decision problem is NP-hard.

math.CO↗

arXiv2025

Thrackles on nonplanar surfaces

A thrackle is a drawing of a graph on a surface such that (i) adjacent edges only intersect at their common vertex; and (ii) nonadjacent edges intersect at exactly one point, at which they cross. Conway conjectured that if a graph with $n$ vertices and $m$ edges can be thrackled on the plane, then $m\le n$. Conway's conjecture remains open; the best bound known is that $m\le 1.393n$. Cairns and Nikolayevsky extended this conjecture to the orientable surface $S_g$ of genus $g > 0$, claiming that if a graph with $n$ vertices and $m$ edges has a thrackle on $S_g$, then $m \le n + 2g$. We disprove this conjecture. In stark contrast with the planar case, we show that for each $g>0$ there is a connected graph with $n$ vertices and $2n + 2g -8$ edges that can be thrackled on $S_g$. This leaves relatively little room for further progress involving thrackles on orientable surfaces, as every connected graph with $n$ vertices and $m$ edges that can be thrackled on $S_g$ satisfies that $m \le 2n + 4g - 2$. We prove a similar result for nonorientable surfaces. We also derive nontrivial upper and lower bounds on the minimum $g$ such that $K_{m,n}$ and $K_n$ can be thrackled on $S_g$.

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arXiv2023

Drawings of Complete Multipartite Graphs Up to Triangle Flips

For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph $K_n$ with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on $n$ vertices is bounded by $O(n^{16})$. The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the sense that having the same ERS does not remain sufficient when removing or adding very few edges.

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arXiv2022

Improved enumeration of simple topological graphs

A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2^O(n^2 log(m/n)), and at most 2^O(mn^{1/2} log n) if m < n^{3/2}. As a consequence we obtain a new upper bound 2^O(n^{3/2} log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2^{n^2 alpha(n)^O(1)}, using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2^{m^2+O(mn)} and at least 2^Omega(m^2) for graphs with m > (6+epsilon)n.

math.CO↗

arXiv2022

Hanani-Tutte for approximating maps of graphs

We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing if a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise non-intersecting "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.

cs.CG↗

arXiv2022

Spiraling and Folding: The Topological View

For every $n$, we construct two curves in the plane that intersect at least $n$ times and do not form spirals. The construction is in three stages: we first exhibit closed curves on the torus that do not form double spirals, then arcs on the torus that do not form spirals, and finally pairs of planar arcs that do not form spirals. These curves provide a counterexample to a proof of Pach and Tóth concerning string graphs.

math.CO↗

arXiv2022

On Crossing-Families in Planar Point Sets

A $k$-crossing family in a point set $S$ in general position is a set of $k$ segments spanned by points of $S$ such that all $k$ segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of $n$ points which do not contain a crossing family of size larger than $8\lceil \frac{n}{41} \rceil$. Both results improve the previously best known bounds.

cs.CG↗

arXiv2022

The $\mathbb{Z}_2$-genus of Kuratowski minors

A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The $\mathbb{Z}_2$-genus of a graph $G$ is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. An unpublished result by Robertson and Seymour implies that for every $t$, every graph of sufficiently large genus contains as a minor a projective $t\times t$ grid or one of the following so-called $t$-Kuratowski graphs: $K_{3,t}$, or $t$ copies of $K_5$ or $K_{3,3}$ sharing at most two common vertices. We show that the $\mathbb{Z}_2$-genus of graphs in these families is unbounded in $t$; in fact, equal to their genus. Together, this implies that the genus of a graph is bounded from above by a function of its $\mathbb{Z}_2$-genus, solving a problem posed by Schaefer and Štefankovič, and giving an approximate version of the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous result for Euler genus and Euler $\mathbb{Z}_2$-genus of graphs.

math.CO↗

arXiv2020

Minimal Representations of Order Types by Geometric Graphs

In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which prevent the order type from changing under continuous motion of vertices. That is, in the geometric graph on S whose edges are the exit edges, in order to change the order type of S, at least one vertex needs to move across an exit edge. Exit edges have a natural dual characterization, which allows us to efficiently compute them and to bound their number.

math.CO↗

arXiv2020

A superlinear lower bound on the number of 5-holes

Let $P$ be a finite set of points in the plane in general position, that is, no three points of $P$ are on a common line. We say that a set $H$ of five points from $P$ is a $5$-hole in $P$ if $H$ is the vertex set of a convex $5$-gon containing no other points of $P$. For a positive integer $n$, let $h_5(n)$ be the minimum number of 5-holes among all sets of $n$ points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for $h_5(n)$ have been of order $Ω(n)$ and $O(n^2)$, respectively. We show that $h_5(n) = Ω(n\log^{4/5}{n})$, obtaining the first superlinear lower bound on $h_5(n)$. The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set $P$ of points in the plane in general position is partitioned by a line $\ell$ into two subsets, each of size at least 5 and not in convex position, then $\ell$ intersects the convex hull of some 5-hole in $P$. The proof of this result is computer-assisted.

math.CO↗

arXiv2019

Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi, Pach, Tóth, and Valtr.

math.CO↗

arXiv2019

Simple realizability of complete abstract topological graphs simplified

An abstract topological graph (briefly an AT-graph) is a pair $A=(G,\mathcal{X})$ where $G=(V,E)$ is a graph and $\mathcal{X}\subseteq {E \choose 2}$ is a set of pairs of its edges. The AT-graph $A$ is simply realizable if $G$ can be drawn in the plane so that each pair of edges from $\mathcal{X}$ crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent $\mathbb{Z}_2$-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.

math.CO↗

arXiv2019

Better upper bounds on the Füredi-Hajnal limits of permutations

A binary matrix is a matrix with entries from the set $\{0,1\}$. We say that a binary matrix $A$ contains a binary matrix $S$ if $S$ can be obtained from $A$ by removal of some rows, some columns, and changing some $1$-entries to $0$-entries. If $A$ does not contain $S$, we say that $A$ avoids $S$. A $k$-permutation matrix $P$ is a binary $k \times k$ matrix with exactly one $1$-entry in every row and one $1$-entry in every column. The Füredi-Hajnal conjecture, proved by Marcus and Tardos, states that for every permutation matrix $P$, there is a constant $c_P$ such that for every $n \in \mathbb{N}$, every $n \times n$ binary matrix $A$ with at least $c_P n$ $1$-entries contains $P$. We show that $c_P \le 2^{O(k^{2/3}\log^{7/3}k / (\log\log k)^{1/3})}$ asymptotically almost surely for a random $k$-permutation matrix $P$. We also show that $c_P \le 2^{(4+o(1))k}$ for every $k$-permutation matrix $P$, improving the constant in the exponent of a recent upper bound on $c_P$ by Fox. Moreover, we improve the upper bound on $c_P$ in terms of the Stanley-Wilf limit $s_P$ to $c_P \le O\big(s_P^{2.75} \log s_P\big)$. We also consider a higher-dimensional generalization of the Stanley-Wilf conjecture about the number of $d$-dimensional $n$-permutation matrices avoiding a fixed $d$-dimensional $k$-permutation matrix, and prove almost matching upper and lower bounds of the form $(2^k)^{O(n)} \cdot (n!)^{d-1-1/(d-1)}$ and $n^{-O(k)} k^{Ω(n)} \cdot (n!)^{d-1-1/(d-1)}$, respectively.

math.CO↗

arXiv2019

On the growth of the Möbius function of permutations

We study the values of the Möbius function $μ$ of intervals in the containment poset of permutations. We construct a sequence of permutations $π_n$ of size $2n-2$ for which $μ(1,π_n)$ is given by a polynomial in $n$ of degree 7. This construction provides the fastest known growth of $|μ(1,π)|$ in terms of $|π|$, improving a previous quadratic bound by Smith. Our approach is based on a formula expressing the Möbius function of an arbitrary permutation interval $[α,β]$ in terms of the number of embeddings of the elements of the interval into $β$.

math.CO↗

arXiv2019

Z_2-genus of graphs and minimum rank of partial symmetric matrices

The \emph{genus} $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. A drawing of a graph on a surface is \emph{independently even} if every pair of nonadjacent edges in the drawing crosses an even number of times. The \emph{$\mathbb{Z}_2$-genus} of a graph $G$, denoted by $\mathrm{g}_0(G)$, is the minimum $g$ such that $G$ has an independently even drawing on $M_g$. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Štefankovič proved that the $\mathbb{Z}_2$-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If $G=G_1\cup G_2$, $G_1$ and $G_2$ intersect in two vertices $u$ and $v$, and $G-u-v$ has $k$ connected components (among which we count the edge $uv$ if present), then $|\mathrm{g}_0(G)-(\mathrm{g}_0(G_1)+\mathrm{g}_0(G_2))|\le k+1$. For complete bipartite graphs $K_{m,n}$, with $n\ge m\ge 3$, we prove that $\frac{\mathrm{g}_0(K_{m,n})}{\mathrm{g}(K_{m,n})}=1-O(\frac{1}{n})$. Similar results are proved also for the Euler $\mathbb{Z}_2$-genus. We express the $\mathbb{Z}_2$-genus of a graph using the minimum rank of partial symmetric matrices over $\mathbb{Z}_2$; a problem that might be of independent interest.

math.CO↗

arXiv2018

The hamburger theorem

We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $μ_1,μ_2, \dots, μ_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $ω_i=μ_i(\mathbb{R}^d)$ for $i\in [d+1]$, $ω=\min\{ω_i; i\in [d+1]\}$ and assume that $\sum_{j=1}^{d+1} ω_j=1$. Assume that $ω_i \le 1/d$ for every $i\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $μ_i(H) \le (\sum_{j=1}^{d+1} μ_j(H))/d$ for every $i \in [d+1]$ and $\sum_{j=1}^{d+1} μ_j(H) \ge \min(1/2, 1-dω) \ge 1/(d+1)$. As a consequence we obtain that every $(d+1)$-colored set of $nd$ points in $\mathbb{R}^d$ such that no color is used for more than $n$ points can be partitioned into $n$ disjoint rainbow $(d-1)$-dimensional simplices.

math.MG↗

arXiv2018

Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4

We find a graph of genus $5$ and its drawing on the orientable surface of genus $4$ with every pair of independent edges crossing an even number of times. This shows that the strong Hanani-Tutte theorem cannot be extended to the orientable surface of genus $4$. As a base step in the construction we use a counterexample to an extension of the unified Hanani-Tutte theorem on the torus.

math.CO↗

arXiv2018

Zeros of the Möbius function of permutations

We show that if a permutation $π$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $μ[π]$ of the interval $[1,π]$ is zero. As a consequence, we show that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^2\ge 0.3995$. This is the first result determining the value of $μ[1,π]$ for an asymptotically positive proportion of permutations $π$. We also show that if a permutation $ϕ$ can be expressed as a direct sum of the form $α\oplus 1 \oplus β$, then any permutation $π$ containing an interval order-isomorphic to $ϕ$ has $μ[1, π]=0$; we deduce this from a more general result showing that $μ[σ, π]=0$ whenever $π$ contains an interval of a certain form. Finally, we show that if a permutation $π$ contains intervals isomorphic to certain pairs of permutations, or to certain permutations of length six, then $μ[1, π] = 0$.

math.CO↗