Radius-two coloring of cubic 2-edge-connected graphs

Construct, for every cubic 2-edge-connected graph, a red-blue vertex coloring in which every monochromatic component has radius at least 1 and at most 2.

Background

The paper establishes that every cubic 2-edge-connected graph has a red-blue coloring whose monochromatic components have radius between 1 and 6. It notes that the Petersen graph forces the upper radius bound to exceed 1, and reports that the proposed radius-two statement has been verified for graphs with up to 18 vertices. The conjecture asks whether radius 2 always suffices.

References

If $G$ is a cubic $2$-edge-connected graph, then there is a red-blue coloring of the vertices of $G$ such that every monochromatic component has radius at least $1$ and at most $2$. This is true up to 18 vertices.

— Two-coloring cubic graphs with small monochromatic components, but without singletons  (2609.30893 - Barát et al., 25 Sep 2026) in Section 2, Conjecture 1 (labeled Conjecture \ref{conj})