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

TL;DR

This paper investigates a specific vertex coloring problem in subcubic graphs related to a conjecture by Thomassen, providing counterexamples and proving the existence of such colorings in certain graph classes.

Contribution

It refutes Thomassen's conjecture with a counterexample and establishes the existence of crumby colorings in outerplanar graphs, subdivisions of $K_4$, and genuine subdivisions of subcubic graphs.

Findings

Counterexample refutes the conjecture for some graphs.

Outerplanar graphs admit crumby colorings.

Genuine subdivisions of subcubic graphs admit crumby colorings.

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, 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 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.