Thomassen’s chord conjecture for longest cycles

Prove that every longest cycle of every 3-connected graph has a chord, thereby resolving the general case of Thomassen’s conjecture.

Background

The paper studies when longest cycles in highly connected graphs must contain chords. A chord is an edge joining two nonconsecutive vertices of a cycle. Thomassen conjectured that every longest cycle in a 3-connected graph has such an edge.

The paper notes that the conjecture has been proved for several important graph classes, including cubic graphs, certain planar graphs, and graphs with sufficiently large minimum degree, but that the general statement remains unresolved. The results in the paper establish further restricted cases involving longest cycles required to pass through specified vertices or edges.