arXiv subjects

Dániel Gábor Simon

Publications and source records attributed to Dániel Gábor Simon.

5 recordsLinked to original sources

arXiv2026

Conway's thrackle conjecture: a new upper bound of $\frac43n$

A thrackle is a drawing of a graph in the plane in which every pair of edges meets exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that every thrackle has at most as many edges as vertices. We prove that every thrackle on $n\ge4$ vertices has at most $\left\lfloor4(n-1)/3\right\rfloor$ edges. The proof combines Pammer's computer-assisted verification of the conjecture for graphs on at most twelve vertices with a discharging argument in suitable plane and projective-plane embeddings.

math.CO↗

arXiv2026

Peeling sequences: a directional method for the three-block construction

A peeling sequence of a finite planar point set is an ordering of its points, such that when the points get removed in this order, each removed point is a vertex of the current convex hull of the set. Write $g(S)$ for the number of such sequences, and $g(n)$ for the minimum of $g(S)$ over $n$-point sets in general position. We present a method which can be used to prove better upper bounds on the previously analysed recursive 3-block constructions $S_n$. In fact, we prove $g(n)\leq g(S_n)=O(6.57^n)$, using directional restrictions and a weighted prefix-tree form of the Kraft--McMillan inequality. From the other direction, we show that the same construction cannot attain the current best known lower bound of $c\cdot3^n$. In fact, $g(S_n)\geq 5.22^n$ when $n$ is a large enough power of $3$. Improving the general lower bound on $g(n)$ is still an open question.

math.CO↗

arXiv2026

Further analysis of Peeling Sequences

Let $P\subset \R^2$ be a set of $n$ points in general position. A peeling sequence of $P$ is a list of its points, such that if we remove the points from $P$ in that order, we always remove the next point from the convex hull of the remainder of $P$. Using the methodology of Dumitrescu and Tóth \cite{Dumitrescu}, with a more careful analysis, we improve the upper bound on the minimum number of peeling sequences for an $n$ point set in the plane from $12.29^n/100$ to $9.78^n/500$.

math.CO↗

arXiv2024

The minimum number of peeling sequences of a point set

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let $g_d(P)$ denote the number of different removal orders we can attain while erasing all points of $P$ this way, and let $g_d(n)$ be the \emph{minimum} of $g_d(P)$ over all $n$-element point sets $P\subset \mathbb{R}^d$. Dumitrescu and Tóth showed that $g_d(n)=(d+1)^{(d+1)^2n}$. We substantially improve their bound, by proving that $g_d(n)= O((d+d\ln{(d)})^{(2+\frac{(d-1)}{\lfloor d\ln{d}\rfloor})n})$. It follows that, for any $ε>0$, there exist sufficiently high dimensional point sets $P\subset \mathbb{R}^d$ with $g_d(P)\leq O(d^{(2+ε)n})$. This almost closes the gap between the upper bound and the best-known lower bound $(d+1)^n$ for large values of $d$.

math.CO↗

arXiv2024

Maximal line-free sets in $\mathbb{F}_p^n$

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and therefore sets containing no full line. A~trivial lower bound $r_p(\mathbb{F}_p^n)\geq(p-1)^n$ is achieved by a hypercube of side length $p-1$ and it is known that equality holds for $n\in\{1,2\}$. We will however show that $r_p(\mathbb{F}_p^3)\geq (p-1)^3+p-2\sqrt{p}$, which is the first improvement in the three dimensional case that is increasing in $p$. We will also give the upper bound $r_p(\mathbb{F}_p^{3})\leq p^3-2p^2-(\sqrt{2}-1)p+2$ as well as generalizations for higher dimensions. Finally we present some bounds for individual $p$ and $n$, in particular $r_5(\mathbb{F}_5^{3})\geq 70$ and $r_7(\mathbb{F}_7^{3})\geq 225$ which can be used to give the asymptotic lower bound $4.121^n$ for $r_5(\mathbb{F}_5^{n})$ and $6.082^n$ for $r_7(\mathbb{F}_7^{n})$.

math.CO↗