Counterexamples to Thomassen's Conjecture on Decomposition of Cubic Graphs

@article{Bellitto2021CounterexamplesTT,
  title={Counterexamples to Thomassen's Conjecture on Decomposition of Cubic Graphs},
  author={Thomas Bellitto and Tereza Klimosov{\'a} and Martin Merker and Marcin Witkowski and Yelena Yuditsky},
  journal={Graphs Comb.},
  year={2021},
  volume={37},
  pages={2595-2599}
}
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices. 

Figures from this paper

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

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