# Decomposition of cubic graphs related to Wegner's conjecture

## TL;DR

This paper proves Thomassen's conjecture for Generalized Petersen graphs and confirms the existence of such colorings for all subcubic trees, advancing understanding of vertex colorings in cubic graphs.

## Contribution

The paper verifies Thomassen's conjecture for a specific class of cubic graphs and extends the idea to all subcubic trees, providing new insights into graph colorings.

## Key findings

- Thomassen's conjecture holds for Generalized Petersen graphs.
- Colorings with the specified properties exist for all subcubic trees.
- Potential extension of the coloring concept to all subcubic graphs.

## 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 $1$ (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. We prove the conjecture for Generalized Petersen graphs. We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.