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Leo Versteegen

Publications and source records attributed to Leo Versteegen.

At least 19 recordsLinked to original sources
arXiv2025

Embedding trees using minimum and maximum degree conditions

A variant of the Erd\H{o}s-S\'os conjecture, posed by Havet, Reed, Stein and Wood, states that every graph with minimum degree at least $\lfloor 2k/3 \rfloor$ and maximum degree at least $k$ contains a copy of every tree with $k$ edges. Both degree bounds are best possible. We confirm this conjecture for large trees with bounded maximum degree, by proving that for all $\Delta\in \mathbb{N}$ and sufficiently large $k\in \mathbb{N}$, every graph $G$ with $\delta(G)\geq \lfloor 2k/3 \rfloor$ and $\Delta(G)\geq k$ contains a copy of every tree $T$ with $k$ edges and $\Delta(T)\leq \Delta$. We also prove similar results where alternative degree conditions are considered. For the same class of trees, this verifies exactly a related conjecture of Besomi, Pavez-Sign\'e and Stein, and provides asymptotic confirmations of two others.

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arXiv2025

Relative Tur\'an densities for ordered graphs: all and nothing

Reiher, R\"odl, Sales, and Schacht initiated the study of relative Tur\'an densities of ordered graphs and showed that it is more subtle and interesting than the unordered case. For an ordered graph $F$, its relative Tur\'an density, $\rho_{<}(F)$, is the greatest $\alpha$ such that every ordered graph $G$ has an $F$-free subgraph with at least $\alpha e(G)$ edges. This paper contains two main results about relative Tur\'an densities. First, we find a family of host graphs that is optimal for all $F$. Second, we characterise the ordered graphs with zero relative Tur\'an density: precisely those with no monotone path of length two.

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arXiv2025

The sandglass conjecture beyond cancellative pairs

The sandglass conjecture, posed by Simonyi, states that if a pair $(A, B)$ of families of subsets of $[n]$ is recovering then $|A| |B| \leq 2^n$. We improve the best known upper bound to $|A| |B| \leq 2.2543^n$. To do this we overcome a significant barrier by exponentially separating the upper bounds on recovering pairs from cancellative pairs, a related notion.

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arXiv2024

On norming systems of linear equations

A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two.

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arXiv2024

A proof of a conjecture of Erd\H{o}s and Gy\'{a}rf\'{a}s on monochromatic path covers

In 1995, Erd\H{o}s and Gy\'{a}rf\'{a}s proved that in every $2$-edge-coloured complete graph on $n$ vertices, there exists a collection of $2\sqrt{n}$ monochromatic paths, all of the same colour, which cover the entire vertex set. They conjectured that it is possible to replace $2\sqrt{n}$ by $\sqrt{n}$. We prove this to be true for all sufficiently large $n$.

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arXiv2024

Packing and finding paths in sparse random graphs

Let $G\sim G(n,p)$ be a (hidden) Erd\H{o}s-R\'enyi random graph with $p=(1+ \varepsilon)/n$ for some fixed constant $ \varepsilon >0$. Ferber, Krivelevich, Sudakov, and Vieira showed that to reveal a path of length $\ell=\Omega\left(\frac{\log(1/ \varepsilon)}{ \varepsilon}\right)$ in $G$ with high probability, one must query the adjacency of $\Omega\left(\frac{\ell}{p \varepsilon\log(1/ \varepsilon)}\right)$ pairs of vertices in $G$, where each query may depend on the outcome of all previous queries. Their result is tight up to the factor of $\log(1/ \varepsilon)$ in both $\ell$ and the number of queries, and they conjectured that this factor could be removed. We confirm their conjecture. The main ingredient in our proof is a result about path-packings in random labelled trees of independent interest. Using this, we also give a partial answer to a related question of Ferber, Krivelevich, Sudakov, and Vieira. Namely, we show that when $\ell=o\left((t/\log t)^{1/3}\right)$, the maximum number of vertices covered by edge-disjoint paths of length at least $\ell$ in a random labelled tree of size $t$ is $\Theta(t/\ell)$ with high probability.

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arXiv2024

An exploration of the balance game

The balance game is played on a graph $G$ by two players, Admirable (A) and Impish (I), who take turns selecting unlabeled vertices of $G$. Admirable labels the selected vertices by $0$ and Impish by $1$, and the resulting label on any edge is the sum modulo $2$ of the labels of the vertices incident to that edge. Let $e_0$ and $e_1$ denote the number of edges labeled by $0$ and $1$ after all the vertices are labeled. The discrepancy in the balance game is defined as $d = e_1 - e_0$. The two players have opposite goals: Admirable attempts to minimize the discrepancy $d$ while Impish attempts to maximize $d$. When (A) makes the first move in the game, the (A)-start game balance number, $b^A_g(G)$, is the value of $d$ when both players play optimally, and when (I) makes the first move in the game, the (I)-start game balance number, $b^I_g(G)$, is the value of $d$ when both players play optimally. Among other results, we show that if $G$ has order $n$, then $-\log_2(n) \le b^A_g(G) \le \frac{n}{2}$ if $n$ is even and $0 \le b^A_g(G) \le \frac{n}{2} + \log_2(n)$ if $n$ is odd. Moreover we show that $b^A_g(G) + b^I_g(\overline{G}) = \lfloor n/2 \rfloor$.

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arXiv2024

Around the positive graph conjecture

A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if it is symmetric, in the sense that it is formed by gluing two copies of some subgraph along an independent set. We prove several results relating to this conjecture. First, we make progress towards the conjecture itself by showing that any connected positive graph must have a vertex of even degree. We then make use of this result to identify some new counterexamples to the analogue of Sidorenko's conjecture for hypergraphs. In particular, we show that, for $r$ odd, every $r$-uniform tight cycle is a counterexample, generalising a recent result of Conlon, Lee and Sidorenko that dealt with the case $r=3$. Finally, we relate the positive graph conjecture to the emerging study of graph codes by showing that any positive graph has vanishing graph code density, thereby improving a result of Alon who proved the same result for symmetric graphs. Our proofs make use of a variety of tools and techniques, including the properties of independence polynomials, hypergraph quasirandomness and discrete Fourier analysis.

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arXiv2023

Upper bounds for linear graph codes

A linear graph code is a family $\mathcal{C}$ of graphs on $n$ vertices with the property that the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In this article, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph $H$. In particular, we show that if $H$ has an even number of edges, the size of the code is $O(2^{\binom{n}{2}}/\log n)$, making progress on a question of Alon. Furthermore, we show that for almost all graphs $H$ with an even number of edges, there exists $\varepsilon_H>0$ such that the size of a linear graph code without a copy of $H$ is at most $2^{\binom{n}{2}}/n^{\varepsilon_H}$.

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arXiv2023

Strongly common graphs with odd girth are cycles

A graph $H$ is called strongly common if for every coloring $\phi$ of $K_n$ with two colors, the number of monochromatic copies of $H$ is at least the number of monochromatic copies of $H$ in a random coloring of $K_n$ with the same density of color classes as $\phi$. In this note we prove that if a graph has odd girth but is not a cycle, then it is not strongly common. This answers a question of Chen and Ma.

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arXiv2023

On interval colourings of graphs

An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\colon E\to \mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\mathbb{Z}$. The interval thickness $\theta(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $\theta(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \frac{\log n}{\log \log n} \leq \theta(n) \leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan. In the second part of the paper, we confirm a conjecture of Axenovich that the maximum number of colours used in an interval colouring of a planar graph on $n$ vertices is at most $3n/2-2$.

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arXiv2023

A note on interval colourings of graphs

A graph is said to be interval colourable if it admits a proper edge-colouring using palette $\mathbb{N}$ in which the set of colours incident to each vertex is an interval. The interval colouring thickness of a graph $G$ is the minimum $k$ such that $G$ can be edge-decomposed into $k$ interval colourable graphs. We show that $\theta(n)$, the maximum interval colouring thickness of an $n$-vertex graph, satisfies $\theta(n) =\Omega(\log(n)/\log\log(n))$ and $\theta(n)\leq n^{5/6+o(1)}$, which improves on the trivial lower bound and an upper bound of the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an $n$-vertex planar graph uses at most $3n/2-2$ colours.

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arXiv2023

The structure of large sum-free sets in $\mathbb{F}_p^n$

A set $A\subset \mathbb{F}_p^n$ is sum-free if $A+A$ does not intersect $A$. If $p\equiv 2 \mod 3$, the maximal size of a sum-free in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. We show that if a sum-free set $A\subset \mathbb{F}_p^n$ has size at least $p^n/3-p^{n-1}/6+p^{n-2}$, then there exists subspace $V<\mathbb{F}_p^n$ of co-dimension 1 such that $A$ is contained in $(p+1)/3$ cosets of $V$. For $p=5$ specifically, we show the stronger result that every sum-free set of size larger than $1.2\cdot 5^{n-1}$ has this property, thus improving on a recent theorem of Lev.

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arXiv2023

Progress towards the 1/2-Conjecture for the domination game

The domination game is played on a graph $G$ by two players, Dominator and Staller, who alternate in selecting vertices until each vertex in the graph $G$ is contained in the closed neighbourhood of the set of selected vertices. Dominator's aim is to reach this state in as few moves as possible, whereas Staller wants the game to last as long as possible. In this paper, we prove that if $G$ has $n$ vertices and minimum degree at least 2, then Dominator has a strategy to finish the domination game on $G$ within $10n/17+1/17$ moves, thus making progress towards a conjecture by Bujt{á}s, Iršič and Klavžar.

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arXiv2023

A proof of the 3/5-conjecture in the domination game

The domination game is an optimization game played by two players, Dominator and Staller, who alternately select vertices in a graph $G$. A vertex is said to be dominated if it has been selected or is adjacent to a selected vertex. Each selected vertex must strictly increase the number of dominated vertices at the time of its selection, and the game ends once every vertex in $G$ is dominated. Dominator aims to keep the game as short as possible, while Staller tries to achieve the opposite. In this article, we prove that for any graph $G$ on $n$ vertices, Dominator has a strategy to end the game in at most $3n/5$ moves, which was conjectured by Kinnersley, West and Zamani.

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arXiv2022

A proof of the 3/4 conjecture for the total domination game

In this paper we describe a strategy for Dominator that finishes the total domination game in at most $3/4n$ moves for every graph $G$ on $n$ vertices without any isolated vertices or edges, confirming the 3/4-conjecture for the total domination game made by Henning, Klav{ž}ar, and Rall.

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arXiv2022

On the number of minimum dominating sets and total dominating sets in forests

We show that the maximum number of minimum dominating sets of a forest with domination number $\gamma$ is at most $\sqrt{5}^{\gamma}$ and construct for each $\gamma$ a tree with domination number $\gamma$ that has more than $\frac{2}{5}\sqrt{5}^{\gamma}$ minimum dominating sets. Furthermore, we disprove a conjecture about the number of minimum total dominating sets in forests by Henning, Mohr and Rautenbach.

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