Tools for parsimonious edge-colouring of graphs with maximum degree three

TL;DR

This paper explores structural properties of $oldsymbol{ ext{delta}}$-minimum edge-colourings in graphs with maximum degree three, providing tools that facilitate efficient edge-colouring in such graphs.

Contribution

It introduces structural insights into $oldsymbol{ ext{delta}}$-minimum edge-colourings, aiding future research and applications in graph colouring.

Findings

Identifies key structural properties of $oldsymbol{ ext{delta}}$-minimum edge-colourings.

Provides auxiliary tools for edge-colouring algorithms.

Supports further research in graph colouring with maximum degree three.

Abstract

The notion of a $\delta$-minimum edge-colouring was introduced by J-L. Fouquet (in his french PhD Thesis \cite{FouPhD}). Here we present some structural properties of $\delta$-minimum edge-colourings, partially taken from the above thesis. The paper serves as an auxiliary tool for another paper submitted by the authors to Graphs and Combinatorics.