Subcubic $K_4$-minor-free graphs without crumby colorings

TL;DR

This paper disproves a conjecture that all $K_4$-minor-free graphs admit a specific type of vertex coloring called crumby coloring, by providing counterexamples within treewidth two graphs.

Contribution

It provides the first known counterexamples to the conjecture that all $K_4$-minor-free graphs have crumby colorings, showing obstructions occur at treewidth two.

Findings

Counterexample with an 18-vertex partial 2-tree disproves the conjecture.

Counterexample with a 40-vertex 2-connected partial 2-tree disproves the 2-connected version.

Obstructions to crumby colorability occur within treewidth two.

Abstract

Motivated by Wegner's conjecture on squares of planar graphs, Thomassen conjectured that every 3-connected cubic graph on at least eight vertices admits a red-blue vertex coloring in which the blue subgraph has maximum degree at most 1, while the red subgraph has minimum degree at least 1 and contains no $P_4$. Such colorings are now called crumby colorings. Although this conjecture was disproved in general by Bellitto, Klimo\v{s}ov\'a, Merker, Witkowski and Yuditsky, positive results of Bar\'at, Bl\'azsik and Dam\'asdi led them, in the same subcubic setting, to conjecture that every $K_4$-minor-free graph admits a crumby coloring. We disprove this conjecture with a connected subcubic partial 2-tree on 18 vertices. We also disprove its natural 2-connected version with a 2-connected subcubic partial 2-tree on 40 vertices with no crumby coloring. Consequently, the obstruction to crumby colorability already occurs within treewidth two, even under 2-connectivity.