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Anusch Taraz

Publications and source records attributed to Anusch Taraz.

At least 19 recordsLinked to original sources
arXiv2026

On Ramsey-type problems for paths and cycles with few colour changes

In 1967, Gerencser and Gy\'arf\'as determined the exact values of the two-colour Ramsey numbers of paths. In a footnote, they made the following observation: Every $2$-edge-coloured complete graph contains a Hamilton path with at most one colour change. Later, this led to a challenging and still wide open conjecture about covering edge-coloured complete graphs with monochromatic paths. Inspired by the original statement, we study paths and cycles with few colour changes in $3$-edge-coloured complete graphs. For this, we introduce a new Ramsey-type parameter: For $q,k \in \mathbb{N}$ and a graph $G$, let $R_q^k(G)$ denote the smallest $N \in \mathbb{N}$ such that every $q$-edge-coloured complete graph on $N$ vertices contains a copy of $G$ with at most $k$ vertices that are incident to edges in $G$ of different colours. For paths, we show that $R_3^1(P_n) = \frac{3n}{2} + O(1)$, and for even cycles, we show that $R_3^2(C_n) = \frac{3n}{2} + o(n)$.

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arXiv2026

The density of graphs with no $\ell$-path connecting equal-degree vertices: a short proof

Addressing a question posed by Chen and Ma from an asymptotic point of view, we present a short proof for the edge density needed to guarantee that two vertices of the same degree are connected by a path of a fixed length. In particular, we show that for any sufficiently large graph, a density of at least $1/2+o(1)$ enforces the existence of two such vertices. This bound is tight for paths of odd length.

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arXiv2024

Randomized algorithms to generate hypergraphs with given degree sequences

The question whether there exists a hypergraph whose degrees are equal to a given sequence of integers is a well-known reconstruction problem in graph theory, which is motivated by discrete tomography. In this paper we approach the problem by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.

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arXiv2021

The tree packing conjecture for trees of almost linear maximum degree

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result actually allows to replace the host graph $K_n$ by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices.

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arXiv2020

On the size-Ramsey number of grid graphs

The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove that the size-Ramsey number of the grid graph on $n\times n$ vertices is bounded from above by $n^{3+o(1)}$.

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arXiv2020

The Bandwidth Theorem in Sparse Graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for $p\gg \big(\tfrac{\log n}{n}\big)^{1/Δ}$ asymptotically almost surely each spanning subgraph $G$ of $G(n,p)$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for $H$ with small degeneracy, which in particular imply a resilience result in $G(n,p)$ with respect to the containment of spanning bounded degree trees for $p\gg \big(\tfrac{\log n}{n}\big)^{1/3}$.

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arXiv2019

A spanning bandwidth theorem in random graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is proved: for $p\gg (\frac{\log n}{n})^{1/Δ}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $(\frac{k-1}{k}+o(1))pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in $G$ contain many copies of $K_Δ$ then we can drop the restriction on $H$ that $Cp^{-2}$ vertices should not be in triangles.

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arXiv2019

Perfectly packing graphs with bounded degeneracy and many leaves

We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $Ω(n)$ of them have at most $(1-Ω(1))n$ vertices and $Ω(n)$ leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

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arXiv2018

Minimal Ramsey graphs for cyclicity

We study graphs with the property that every edge-colouring admits a monochromatic cycle (the length of which may depend freely on the colouring) and describe those graphs that are minimal with this property. We show that every member in this class reduces recursively to one of the base graphs $K_5-e$ or $K_4\vee K_4$ (two copies of $K_4$ identified at an edge), which implies that an arbitrary $n$-vertex graph with $e(G)\geq 2n-1$ must contain one of those as a minor. We also describe three explicit constructions governing the reverse process. As an application we are able to establish Ramsey infiniteness for each of the three possible chromatic subclasses $χ=2, 3, 4$, the unboundedness of maximum degree within the class as well as Ramsey separability of the family of cycles of length $\leq l$ from any of its proper subfamilies.

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arXiv2018

Logical limit laws for minor-closed classes of graphs

Let $\mathcal G$ be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all {\em connected} graphs in $\mathcal G$ on $n$ vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in $\mathcal G$ on $n$ vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface $S$. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of length $\approx 5\cdot 10^{-6}$. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on $n$ vertices, even in FO.

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arXiv2017

On Minimum Bisection and Related Cut Problems in Trees and Tree-Like Graphs

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider this problem for trees and prove that the minimum bisection width of every tree $T$ on $n$ vertices satisfies $MinBis(T) \leq 8 n Δ(T) / diam(T)$. Second, we generalize this to arbitrary graphs with a given tree decomposition $(T,X)$ and give an upper bound on the minimum bisection width that depends on the structure of $(T,X)$. Moreover, we show that a bisection satisfying our general bound can be computed in time proportional to the encoding length of the tree decomposition when the latter is provided as input.

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arXiv2017

Approximating the Minimum $k$-Section Width in Bounded-Degree Trees with Linear Diameter

Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) Δ(T)$. This implies a polynomial-time constant-factor approximation for the Minimum $k$-Section Problem when restricted to trees with linear diameter and constant maximum degree. Moreover, we extend our results from trees to arbitrary graphs with a given tree decomposition.

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arXiv2016

Counting results for sparse pseudorandom hypergraphs II

We present a variant of a universality result of Rödl [On universality of graphs with uniformly distributed edges, Discrete Math. 59 (1986), no. 1-2, 125-134] for sparse, $3$-uniform hypergraphs contained in strongly jumbled hypergraphs. One of the ingredients of our proof is a counting lemma for fixed hypergraphs in sparse ``pseudorandom'' uniform hypergraphs, which is proved in the companion paper [Counting results for sparse pseudorandom hypergraphs I].

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arXiv2016

Counting results for sparse pseudorandom hypergraphs I

We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].

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arXiv2016

Ramsey numbers for bipartite graphs with small bandwidth

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.

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arXiv2014

Coloring d-Embeddable k-Uniform Hypergraphs

This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and strong) chromatic number of hypergraphs in H(d,k). For example, we can prove that for d>2 there are hypergraphs in H(2d-3,d) on n vertices whose weak chromatic number is Omega(log n/log log n), whereas the weak chromatic number for n-vertex hypergraphs in H(d,d) is bounded by O(n^((d-2)/(d-1))) for d>2.

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arXiv2014

An approximate version of the Tree Packing Conjecture

We prove that for any pair of constants $ε>0$ and $Δ$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $Δ$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+ε)n}$. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.

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