Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.07314

On the number of edges in quasiplanar graphs

János Pach·Gábor Tardos·Ji Zeng
View original source ↗Read source document ↗Export metadata ↓

Abstract

A $k$-quasiplanar graph is a graph drawn in the plane with no $k$ pairwise crossing edges. Pach, Shahrokhi, and Szegedy conjectured that every such graph on $n$ vertices has $O_k(n)$ edges. ChatGPT produced a proof of this conjecture. Here, we present a simplified, self-contained version of that proof.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

János Pach, Gábor Tardos, Ji Zeng. 2026-11-05. On the number of edges in quasiplanar graphs. https://arxiv.org/abs/2611.07314

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Domination in Johnson graphs J(n, 3) for odd n

In 2025 Cornet, Dravec, and Torres determined the domination number $γ(J(n,3))$ of the Johnson graph for every even $n \ge 6$, expressing it as a closed form $ϕ_n$ in terms of Fort-Hedlund covering numbers, and conjectured the same value for odd $n$. We prove this conjecture: $γ(J(n,3)) = ϕ_n$ for every odd $n \ge 7$. A dominating set of $J(n,3)$ is the same as a lotto design $\mathrm{LD}(n,3,3,2)$, and the lotto numbers $L(n,3,3,2)$, and hence $γ(J(n,3))$, were already determined for every $n$ by Bate (1978) and by Brouwer (1981); a short degree argument of Füredi, Székely, and Zubor (1996) gives the same lower bound for all $n \not\equiv 9, 10 \pmod{12}$. The contribution of this paper is therefore a new proof of the odd case, carried out within the bijective framework of Cornet, Dravec, and Torres.

math.CO↗
arXiv2026

HJ numbers revisited

Hales-Jewett theorem is well known. We prove a primitive recursive bound, in fact an $\clE^5$ one, for the density version. In preparation we deal with what was the main induction step in the old proof of Hales-Jewett as a result in its own right with better bounds. Also in improving our bounds.

math.CO↗
arXiv2026

Adjacency-degree algebras and spectral determination of graphs

Let $G$ be a graph with adjacency matrix $A$ and degree matrix $D$. Applying $A$ and $D$ repeatedly to the all-ones vector produces a space $M_G$ of functions on the vertices, and these functions are constant on the orbits of the automorphism group. We call $G$ AD-observable if $M_G$ contains the indicator vector of every orbit, and AD-separable if the functions in $M_G$ separate the orbits. Forests are AD-observable, by McKay's leaf-pruning argument. Our main result is that for every cograph, $M_G$ is spanned by the indicator vectors of the cells of the stable partition of color refinement. Hence a cograph is AD-observable, or equivalently AD-separable, exactly when color refinement finds its orbits, and we describe these cographs by their cotrees. We also show that two cographs with the same number of vertices and the same homomorphism counts from all caterpillars have the same counts from all trees. The main result does not extend to distance-hereditary graphs: we give two such graphs on nine vertices that are AD-separable but not AD-observable.

math.CO↗