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

Abstract

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring \emph{crumby}. Recently, Bellitto, Klimošová, 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 for some important graph classes: outerplanar graphs, $K_4$-minor-free graphs, bipartite graphs. In this regard, we prove that $2$-connected outerplanar graphs, subdivisions of $K_4$ and $1$-subdivisions of cubic graphs admit crumby colorings. A subdivision of $G$ is {\it genuine} if every edge is subdivided at least once. We show that every genuine subdivision of any subcubic graph admits a crumby coloring. We slightly generalise some of these results and formulate a few conjectures.