Chords in longest cycles through a specified edge

Prove that, for every 3-connected graph and every specified edge, each longest cycle containing that edge has a chord.

Background

This conjecture strengthens the global longest-cycle chord problem by requiring the cycle to contain a prescribed edge. It asks whether the chord property persists when longest cycles are optimized within the family of cycles passing through a given local feature.

The paper proves the conjecture when the graph is 3-connected with minimum degree at least 4 and partially verifies it in other settings, including specified small sets in 3-connected planar graphs. The full assertion for arbitrary 3-connected graphs is not established.

References

Conjecture 1.6 (Gu, Jia and Wu [8]) Let G be a 3-connected graph and e be an edge of G. Then among all the cycles containing e, each longest such cycle has a chord. Inspired by Conjecture 1.6, we study longest cycles containing a specified small set and generalize Theorems 1.2-1.5 as follows. Our results also partially verify Conjecture 1.6.

— Chords of longest cycles passing through a specified small set  (2502.10657 - Wu et al., 15 Feb 2025) in Conjecture 1.6, Section 1 (Introduction), p. 3