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Éva Czabarka

Publications and source records attributed to Éva Czabarka.

At least 19 recordsLinked to original sources
arXiv2026

Curious crossing-critical edges

Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we present the following examples: a graph with an edge that is crossed in every optimal drawing of the graph, but the edge is not in any Kuratowski subgraph of the graph; a graph with an edge that is in every Kuratowski subgraph but is not crossed in any optimal drawing of the graph; and a graph with a crossing-critical edge that is not present in any Kuratowski subgraph and is not crossed in any optimal drawing of the graph. Fáry's theorem implies that the Kuratowski subgraphs are the only obstructions to a graph having a crossing-free drawing with all edges drawn as straight lines. The three example graphs given also hold if we restrict drawings to only have straight line edges, and thus also apply to the rectilinear crossing number.

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arXiv2026

On Agreement Subtrees in Multiple Phylogenetic Trees

Snir and Yuster [Discrete Appl. Math. 347 (2026) 160--171] asked for the least number $h(k)$ such that $k$ unrooted binary phylogenetic trees on the same $h(k)$ leaves always share a common quartet. We give a new upper bound for the $k$-tree version of the Maximum Agreement Subtree problem, namely an upper bound for the number of leaves, on which $k$ unrooted binary phylogenetic trees always share a common induced binary subtree on $n$ leaves, which is a four-times iterated exponential function. For $h(k)$, this implies a four-times iterated exponential upper bound. We also set an exponential lower bound for $h(k)$.

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arXiv2025

Tanglegrams with a Unique 1-Crossing-Critical Subtanglegram have Tangle Crossing Number 1

A tanglegram of size n is a graph formed from two rooted binary trees with n leaves each and a perfect matching between their leaf sets. Tanglegrams are used to model co-evolution in various settings. A tanglegram layout is a straight line drawing where the two trees are drawn as plane trees with their leaf-sets on two parallel lines, and only the edges of the matching may cross. The tangle crossing number of a tanglegram is the minimum crossing number among its layouts. It is known that tanglegrams have crossing number at least one precisely when they contain one of two size 4 subtanglegrams, which we refer to as cross-inducing subtanglegrams. We show here that a tanglegram with exactly one cross inducing subtanglegram must have tangle crossing number exactly one, and ask the question whether the tangle-crossing number of tanglegrams with exactly k cross-inducing subtanglegrams is bounded for every k.

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arXiv2023

A Note on the 2-Colored Rectilinear Crossing Number of Random Point Sets in the Unit Square

Let $S$ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of $S$ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that $S$ defines a pair of crossing edges of the same color is equal to $1/4$. This is connected to a recent result of Aichholzer et al. [GD 2019] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halfed. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation $\frac{1}{2}-\frac{7}{50}$ of the total number of crossings.

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arXiv2023

The largest crossing number of tanglegrams

A tanglegram $\cal T$ consists of two rooted binary trees with the same number of leaves, and a perfect matching between the two leaf sets. In a layout, the tanglegrams is drawn with the leaves on two parallel lines, the trees on either side of the strip created by these lines are drawn as plane trees, and the perfect matching is drawn in straight line segments inside the strip. The tanglegram crossing number ${\rm cr}({\cal T})$ of $\cal T$ is the smallest number of crossings of pairs of matching edges, over all possible layouts of $\cal T$. The size of the tanglegram is the number of matching edges, say $n$. An earlier paper showed that the maximum of the tanglegram crossing number of size $n$ tanglegrams is $<\frac{1}{2}\binom{n}{2}$; but is at least $\frac{1}{2}\binom{n}{2}-\frac{n^{3/2}-n}{2}$ for infinitely many $n$. Now we make better bounds: the maximum crossing number of a size $n$ tanglegram is at most $ \frac{1}{2}\binom{n}{2}-\frac{n}{4}$, but for infinitely many $n$, at least $\frac{1}{2}\binom{n}{2}-\frac{n\log_2 n}{4}$. The problem shows analogy with the Unbalancing Lights Problem of Gale and Berlekamp.

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arXiv2022

Proximity in Triangulations and Quadrangulations

Let $ G $ be a connected graph. If $\barσ(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the proximity, $π(G)$, of $G$ is defined as the smallest value of $\barσ(v)$ over all vertices $v$ of $G$. We give upper bounds for the proximity of simple triangulations and quadrangulations of given order and connectivity. We also construct simple triangulations and quadrangulations of given order and connectivity that match the upper bounds asymptotically and are likely optimal.

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arXiv2021

Minimum Wiener Index of Triangulations and Quadrangulations

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple triangulations and quadrangulations with connectivity at least $c$, and provide the extremal structures, which attain those values. Our main tool is setting upper bounds for the maximum degree in highly connected triangulations and quadrangulations.

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arXiv2021

Maximum diameter of $3$- and $4$-colorable graphs

P. Erdős, J. Pach, R. Pollack, and Z. Tuza [J. Combin. Theory, B 47 (1989), 279--285] made conjectures for the maximum diameter of connected graphs without a complete subgraph $K_{k+1}$, which have order $n$ and minimum degree $δ$. Settling a weaker version of a problem, by strengthening the $K_{k+1}$-free condition to $k$-colorable, we solve the problem for $k=3$ and $k=4$ using a unified linear programming duality approach. The case $k=4$ is a substantial simplification of the result of É. Czabarka, P. Dankelmann, and L. A. Székely [Europ. J. Comb., 30 (2009), 1082--1089].

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arXiv2021

Wiener Index and Remoteness in Triangulations and Quadrangulations

Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overlineσ(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overlineσ(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.

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arXiv2020

On the maximum diameter of $k$-colorable graphs

Erdős, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $δ\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}δ + O(1)$ for every $r\ge 2$, if $δ$ is a multiple of $(r-1)(3r+2)$. For every $r>1$ and $δ\ge 2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $δ$ and diameter $\frac{(6r-5)n}{(2r-1)δ+2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $δ>2(r-1)(3r+2)(2r-3)$. The rest of the paper proves positive results under a stronger hypothesis, $k$-colorability, instead of being $K_{k+1}$-free. We show that the diameter of connected $k$-colorable graphs with minimum degree $\geq δ$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}δ+O(1)$, while for $k=3$, it is at most $\frac{57n}{23δ}+O\left(1\right)$.

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arXiv2020

An infinite antichain of planar tanglegrams

Contrary to the expectation arising from the tanglegram Kuratowski theorem of É. Czabarka, L.A. Székely and S. Wagner [SIAM J. Discrete Math. 31(3): 1732--1750, (2017)], we construct an infinite antichain of planar tanglegrams with respect to the induced subtanglegram partial order. R.E. Tarjan, R. Laver, D.A. Spielman and M. Bóna, and possibly others, showed that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain, i.e. there exists an infinite collection of permutations, such that none of them contains another as a pattern. Our construction adds a twist to the construction of Spielman and Bóna [Electr. J. Comb, Vol. 7. N2.]

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arXiv2018

Midrange crossing constants for graphs classes

For positive integers $n$ and $e$, let $κ(n,e)$ be the minimum crossing number (the standard planar crossing number) taken over all graphs with $n$ vertices and at least $e$ edges. Pach, Spencer and Tóth [Discrete and Computational Geometry 24 623--644, (2000)] showed that $κ(n,e) n^2/e^3$ tends to a positive constant (called midrange crossing constant) as $n\to \infty$ and $n \ll e \ll n^2$, proving a conjecture of Erdős and Guy. In this note, we extend their proof to show that the midrange crossing constant exists for graph classes that satisfy a certain set of graph properties. As a corollary, we show that the the midrange crossing constant exists for the family of bipartite graphs. All these results have their analogues for rectilinear crossing numbers.

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arXiv2018

A Size Condition for Diameter Two Orientable Graphs

It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745--756] that for $n\geq 5$ every simple graph of order $n$ and size at least $\binom{n}{2}-n+5$ has an orientation of diameter two. We prove this conjecture and hence determine for every $n\geq 5$ the minimum value of $m$ such that every graph of order $n$ and size $m$ has an orientation of diameter two.

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arXiv2018

Erdős-Szekeres theorem for cyclic permutations

We provide a cyclic permutation analogue of the Erd\H os-Szekeres theorem. In particular, we show that every cyclic permutation of length $(k-1)(\ell-1)+2$ has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $\ell+1$, and show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $\ell+1$.

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arXiv2018

Inducibility of d-ary trees

Imitating a recently introduced invariant of trees, we initiate the study of the inducibility of $d$-ary trees (rooted trees whose vertex outdegrees are bounded from above by $d\geq 2$) with a given number of leaves. We determine the exact inducibility for stars and binary caterpillars. For $T$ in the family of strictly $d$-ary trees (every vertex has $0$ or $d$ children), we prove that the difference between the maximum density of a $d$-ary tree $D$ in $T$ and the inducibility of $D$ is of order $\mathcal{O}(|T|^{-1/2})$ compared to the general case where it is shown that the difference is $\mathcal{O}(|T|^{-1})$ which, in particular, responds positively to an existing conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in $d$-ary trees is independent of $d$. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every $d$.

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arXiv2017

Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $γ(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\leγ(n)<\frac{1}{2}\binom{n}{2}$. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in $O(n^4)$ time. This lower bound may be tight, even for tanglegrams with tangle crossing number $Θ(n^2)$.

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