Space crossing numbers

TL;DR

This paper introduces a new concept of crossing number for graph embeddings in three-dimensional space, establishes a lower bound related to the classical crossing lemma, and provides sharp bounds for pseudo-random graphs.

Contribution

It defines space crossing numbers, derives a near-classical lower bound, and determines sharp bounds for pseudo-random graphs, advancing understanding of graph embeddings in 3D.

Findings

Lower bound on space crossing number almost implies crossing lemma

Sharp bounds on space crossing numbers of pseudo-random graphs

Extension of crossing number concepts to 3D embeddings

Abstract

We define the crossing number for an embedding of a graph G into R^3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the space crossing numbers of pseudo-random graphs.