Proper edge coloring ($e\ne f,e\cap f\ne \varnothing ⟹c(e)\ne c(f)$)

A proper edge colouring gives different labels to any two distinct edges sharing a vertex. We work with finite undirected graphs without self-loops; distinct parallel edges in a multigraph must also have different colours. For a simple graph, whose edges are two-element sets, the condition isEvery colour class $M_i=\{e\in E(G):c(e)=i\}$ is a matching in a graph. Conversely, a set partition of the edges into matchings in a graph supplies a proper edge colouring. Thus minimizing colours means grouping all the edges into as few matchings in a graph as possible.

The line graph makes the relationship with proper vertex colouring explicit: replace each original edge by a vertex, joining two new vertices exactly when their original edges share an endpoint. A proper edge colouring of $G$ becomes a proper vertex colouring of its line graph $L(G)$, so $\chi \prime (G)=\chi (L(G))$. For a multigraph, retain a distinct new vertex for each parallel edge.^[ref]