# Oriented Colourings of Graphs with Maximum Degree Three and Four

## TL;DR

This paper improves upper bounds on the number of colours needed for oriented colourings of graphs with maximum degrees three and four, advancing understanding of graph orientation colourings.

## Contribution

It provides tighter upper bounds for oriented colourings of graphs with maximum degrees three and four, reducing previous bounds from 11 to 9 and from 80 to 69.

## Key findings

- Graphs with maximum degree three can be oriented with 9 colours.
- Graphs with maximum degree four can be oriented with 69 colours.
- Improved bounds surpass previous known upper limits.

## Abstract

We show that any orientation of a graph with maximum degree three has an oriented 9-colouring, and that any orientation of a graph with maximum degree four has an oriented 69-colouring. These results improve the best known upper bounds of 11 and 80, respectively.