On a conjecture of Pach-Spencer-Tóth for graph crossing numbers

TL;DR

This work resolves the Pach–Spencer–Tóth conjecture by proving an optimal lower bound $cr(G)\ge c'\frac{e^{2+1/\alpha}}{n^{1+1/\alpha}}$ for graphs with $n$ vertices and $e$ edges whose every subgraph satisfies $e(H)\le A\,n(H)^{1+\alpha}$. It also clarifies the role of bisection width in bounding crossings, establishing a sharp $t$-threshold (valid precisely for $0<t\le 2$) in the relation $b(G)=O\big(\sqrt{cr(G)}+(\sum d_i^t)^{1/t}\big)$, and provides a dual result bounding edges from crossing-number information. The authors derive corollaries for $C_{2k}$-free graphs via Bondy–Simonovits, and prove a dual theorem using a minimal-counterexample framework with $(N,\alpha)$-good graphs. Overall, the results deepen the connection between extremal graph structure, topological crossing properties, and related geometric problems.

Abstract

The crossing number of a graph $G$ denotes the minimum number of crossings in any planar drawing of $G$. In this short note, we confirm a long-standing conjecture posed by Pach, Spencer, and Tóth over 25 years ago, establishing an optimal lower bound on the crossing number of graphs that satisfy some monotone properties. Furthermore, we address a related open problem introduced by Pach and Tóth in 2000, which explores the interplay between the crossing number of a graph, its degree sequence, and its bisection width.

Figures (1)

• Figure 1: A drawing of $G$ with $n=5$

Theorems & Definitions (11)

• Conjecture 1.1: Pach-Spencer-Tóth, PST
• Theorem 1.2
• Corollary 1.3
• Theorem 1.5
• Theorem 1.6
• proof : Proof of Theorem \ref{['problem']}
• Remark 2.1
• proof : Proof of Theorem \ref{['main']}
• Theorem 3.1: Bondy-Simonovits, BS
• proof : Proof of Corollary \ref{['C2k']}