Every longest circuit of a 3-connected, $K_{3,3}$-minor free graph has a chord

TL;DR

This paper proves Thomassen's conjecture that every longest circuit in a 3-connected, $K_{3,3}$-minor free graph has a chord, including planar graphs, advancing understanding of circuit structure in such graphs.

Contribution

The paper confirms Thomassen's conjecture for graphs without a $K_{3,3}$ minor, including all planar graphs, providing a significant structural insight.

Findings

Every longest circuit in a 3-connected, $K_{3,3}$-minor free graph has a chord.

The conjecture holds for all planar graphs.

Structural properties of circuits in minor-free graphs are clarified.

Abstract

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs.