On a conjecture of Pach-Spencer-T\'oth for graph crossing numbers

TL;DR

This paper confirms a 25-year-old conjecture by Pach, Spencer, and Toth, establishing an optimal lower bound on the crossing number for graphs with certain properties, and also addresses related open problems involving degree sequences and bisection width.

Contribution

It proves a long-standing conjecture on the crossing number bounds for graphs with monotone properties and explores the relationship between crossing number, degree sequence, and bisection width.

Findings

Confirmed the conjecture with an optimal lower bound.

Linked crossing number bounds to degree sequences and bisection width.

Addressed open problems from Pach and Toth (2000).

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 T\'oth 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 T\'oth in 2000, which explores the interplay between the crossing number of a graph, its degree sequence, and its bisection width.