# Counterexamples to Thomassen's conjecture on decomposition of cubic graphs

## TL;DR

This paper constructs an infinite family of counterexamples disproving Thomassen's conjecture on vertex coloring properties of 3-connected cubic graphs, challenging previous assumptions in graph theory.

## Contribution

The authors provide the first known infinite family of counterexamples to Thomassen's conjecture, demonstrating its invalidity.

## Key findings

- Counterexamples exist for all sufficiently large 3-connected cubic graphs.
- Thomassen's conjecture does not hold universally for graphs with at least 8 vertices.
- The construction method can be applied to generate further counterexamples.

## Abstract

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.