Thomassen’s conjecture on chords of longest cycles

Prove that every longest cycle of every 3-connected graph has a chord.

Background

The paper identifies Thomassen’s conjecture as a long-standing problem concerning the global structure of longest cycles in 3-connected graphs. A chord is an edge joining two nonconsecutive vertices of a cycle.

Several important special cases are known, including cubic graphs, certain planar graphs, graphs of sufficiently large minimum degree, and various embedded or minor-restricted graph classes. The unrestricted 3-connected case remains unresolved, and the results in the paper address related longest-cycle chord questions under additional degree, planarity, connectivity, or prescribed-vertex conditions.

References

Conjecture 1.1 (Thomassen [1],[17]) Every longest cycle of a 3-connected graph has a chord. Although the general conjecture remains unsolved, many partial results have been discovered.

— Chords of longest cycles passing through a specified small set  (2502.10657 - Wu et al., 15 Feb 2025) in Conjecture 1.1, Section 1 (Introduction), p. 2