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Thomassen formulated the following conjecture: Every $3$-connected cubicgraph has a red-blue vertex coloring such that the blue subgraph has maximumdegree at most $1$ (that is, it consists of a matching and some isolatedvertices) and the red subgraph has minimum degree at least $1$ and contains no$3$-edge path. Since all monochromatic components are small in this coloringand there is a certain irregularity, we call such a coloring \emph{crumby}.Recently, Bellitto, Klimo\v{s}ov\'a, Merker, Witkowski and Yuditsky\cite{counter} constructed an infinite family refuting the above conjecture.Their prototype counterexample is $2$-connected, planar, but contains a$K_4$-minor and also a $5$-cycle. This leaves the above conjecture open forsome important graph classes: outerplanar graphs, $K_4$-minor-free graphs,bipartite graphs. In this regard, we prove that $2$-connected outerplanargraphs, subdivisions of $K_4$ and $1$-subdivisions of cubic graphs admit crumbycolorings. A subdivision of $G$ is {\it genuine} if every edge is subdivided atleast once. We show that every genuine subdivision of any subcubic graph admitsa crumby coloring. We slightly generalise some of these results and formulate afew conjectures.

Crumby colorings -- red-blue vertex partition of subcubic graphs regarding a conjecture of Thomassen