Optimal monochromatic-component bound for cubic graphs with perfect matchings

Determine the minimum integer k such that every cubic graph admitting a perfect matching has a red-blue vertex coloring in which every monochromatic component has at most k vertices and no monochromatic component is a singleton.

Background

The paper proves that every cubic graph with a perfect matching has a red-blue coloring whose monochromatic components have between 2 and 12 vertices. A Heawood-graph construction shows that a universal bound cannot be smaller than 7. The unresolved problem is to determine the exact optimal value between these bounds.

References

For any cubic graph $G$ that admits a perfect matching what is the minimum number $k$ for which there always exists a red--blue coloring of $V(G)$ such that the maximum order of a monochromatic component is at most $k$ while there are no monochromatic singletons?

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