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 crumby. Recently, Bellitto, Klimošová, Merker, Witkowski and Yuditsky [2] constructed an infinite family refuting the above conjecture. Their prototype counterexample is 2-connected, planar, but contains a K4-minor and also a 5-cycle. This leaves the above conjecture open for some important graph classes: outerplanar graphs, K4-minor-free graphs, bipartite graphs. In this regard, we prove that 2-connected outerplanar graphs, subdivisions of K4 and 1-subdivisions of cubic graphs admit crumby colorings. A subdivision of G is genuine if every edge is subdivided at least once. We show that every genuine subdivision of any subcubic multigraph admits a crumby coloring. We slightly generalize some of these results and formulate a few conjectures.