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

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.