# Variations on the Petersen colouring conjecture

## TL;DR

This paper investigates a variation of the Petersen colouring conjecture, providing a near-solution by showing that most edges in a bridgeless cubic graph can be coloured with four colours to nearly satisfy the conjecture's conditions.

## Contribution

It introduces a new partial colouring approach for bridgeless cubic graphs that nearly satisfies the Petersen colouring conjecture, with a tight bound proven.

## Key findings

- A 4-colour edge-colouring where at most 80% of edges violate the conjecture's condition.
- The bound is tight and only the Petersen graph reaches this limit.
- A simple discharging method is used for the proof.

## Abstract

The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with $5$ colours such that for every edge $e$, the set of colours assigned to the edges adjacent to $e$ has cardinality either $2$ or $4$, but not $3$. We prove that every bridgeless cubic graph $G$ admits an edge-colouring with $4$ colours such that at most $\frac45\cdot|V(G)|$ edges do not satisfy the above condition. This bound is tight and the Petersen graph is the only connected graph for which the bound cannot be decreased. We obtain such a $4$-edge-colouring by using a carefully chosen subset of edges of a perfect matching, and the analysis relies on a simple discharging procedure with essentially no reductions and very few rules.