Improved generalized crumby bound

Prove that every subcubic graph admits an ℓ-crumby coloring for some integer ℓ<52, and determine whether ℓ=4 suffices.

Background

An ℓ-crumby coloring requires the blue induced subgraph to have maximum degree at most 1, the red induced subgraph to have minimum degree at least 1, and every red path to have at most ℓ edges. The paper proves existence with ℓ=88 and observes that its method can be improved to ℓ=52. The stated problem asks for a further improvement below 52 and specifically questions whether the much smaller value 4 is sufficient.

References

Prove the existence of an $\ell$-crumby coloring for any subcubic graph for some $\ell<52$. Is it true that $\ell=4$ is enough?

— Two-coloring cubic graphs with small monochromatic components, but without singletons  (2609.30893 - Barát et al., 25 Sep 2026) in Section 4, Discussion, third Problem