Linear chord density in minimum-degree-three graphs

Prove or disprove that there exists a constant c>0 such that every graph with minimum degree 3 contains a cycle of length ℓ having at least cℓ chords.

Background

The main results guarantee cycles with many chords under arbitrarily high minimum-degree assumptions. The authors investigate whether an analogous phenomenon can hold already for minimum degree 3, noting that disjoint unions of copies of K_4 rule out an unrestricted statement without additional considerations such as girth. They formulate the concrete unresolved question of whether the number of chords can be linear in the length of a suitable cycle.

References

This leads to the following question: Is there a constant $c>0$ such that every graph with minimum degree 3 contains a cycle of length $\ell$ with at least $c \ell$ chords?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 5, “Concluding remarks and open problems”

Is there a constant $c>0$ such that every graph with minimum degree 3 contains a cycle of length $\ell$ with at least $c \ell$ chords?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 5, “Concluding remarks and open problems”

It is then reasonable to link the number of chords to the length of the host cycle. This leads to the following question: Is there a constant $c>0$ such that every graph with minimum degree 3 contains a cycle of length $\ell$ with at least $c \ell$ chords?

— Lollipops, dense cycles and chords  (2502.04726 - Dvořák et al., 7 Feb 2025) in Section 4, “Concluding remarks and open problems”