Embedding products of graphs into Euclidean spaces

TL;DR

This paper determines the minimal Euclidean dimension needed to embed products of graphs, solving a longstanding problem by linking graph embedding properties to topological link theory.

Contribution

It establishes the minimal embedding dimension for graph products, including proving that certain graph powers cannot embed into lower-dimensional Euclidean spaces, solving Menger's 1929 problem.

Findings

n-th powers of Kuratowsky graphs are not embeddable into 2n-dimensional space

Reduction to Ramsey link theory links embedding problems to topological linking

Provides minimal embedding dimensions for products of graphs

Abstract

For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the 2n-dimensional Euclidean space. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding of L into the (2n-1)-dimensional sphere, where L is the join of n copies of a 4-point set, has a pair of linked (n-1)-dimensional spheres.