Farber's conjecture for planar graphs

TL;DR

This paper proves that planar graphs have the highest possible topological complexity in their ordered configuration spaces, confirming a conjecture of Farber and extending the result to all higher complexities.

Contribution

It establishes the generic maximality of topological complexity for planar graphs and discusses limitations of the approach for non-planar graphs.

Findings

Ordered configuration spaces of planar graphs have maximal topological complexity.

The result confirms Farber's conjecture for planar graphs.

Standard methods fail for non-planar graphs at a fundamental level.

Abstract

We prove that the ordered configuration spaces of planar graphs have the highest possible topological complexity generically, as predicted by a conjecture of Farber. Our argument establishes the same generic maximality for all higher topological complexities. We include some discussion of the non-planar case, demonstrating that the standard approach to the conjecture fails at a fundamental level.