The crumby coloring conjecture for subcubic outerplanar graphs

A crumby coloring is a red-blue vertex coloring in which the blue subgraph has maximum degree at most $1$ and the red subgraph has minimum degree at least $1$ and contains no path with $3$ edges. The outerplanar crumby coloring conjecture. Every outerplanar graph with maximum degree $3$ admits a crumby coloring. The source proves the corresponding statement for $2$-connected subcubic outerplanar graphs and explains that extending it to all subcubic outerplanar graphs requires handling tree-like attachments between blocks. The full conjecture remains open in the source.