Chord conjecture for longest cycles through a linear forest

Prove that, for every k-connected graph with k ≥ 2 and every linear forest subgraph having l edges and t isolated vertices with l + t ≤ k − 2, every longest cycle passing through that linear forest has a chord.

Background

A linear forest is a graph whose components are paths, including isolated vertices. A cycle passes through such a forest when it contains all edges and vertices of the forest.

The conjecture generalizes the specified-edge conjecture by allowing the prescribed subgraph to be an arbitrary sufficiently small linear forest. The size condition l + t ≤ k − 2 is aligned with an earlier theorem guaranteeing a long cycle through the forest, but the stronger assertion that every longest such cycle has a chord is proposed as an unresolved problem.

References

So we propose the following more general conjecture than conjecture 1.6.Conjecture 6.2. Let G be a k-connected graph (k ≥ 2) and let F be a linear forest subgraph of G with l edges and t isolated vertices such that l + t ≤ k − 2. Then every longest cycle of G passing through F has a chord.

— Chords of longest cycles passing through a specified small set  (2502.10657 - Wu et al., 15 Feb 2025) in Conjecture 6.2, Section 6, page 15

It is natural to ask whether more general results than our main theorems are true. So we propose the following more general conjecture than conjecture 1.6. Conjecture 6.2. Let G be a k-connected graph (k ≥ 2) and let F be a linear forest subgraph of G with l edges and t isolated vertices such that l + t ≤ k − 2. Then every longest cycle of G passing through F has a chord.

— Chords of longest cycles passing through a specified small set  (2502.10657 - Wu et al., 15 Feb 2025) in Conjecture 6.2, Section 6 (A new conjecture)