A (proper) $k$-edge colouring of a graph $G(V,E)$ is a function $f:E\to\{1,2,\dots,k\}$ such that adjacent vertices are mapped to different colours; that is, $f(e)\neq f(e')$ if $e$ and $e'$ are incident on the same vertex. It is known that the minimum number of colours required for edge colouring is between $\Delta(G)$ and $\Delta(G)+1$, and yet it is NP-complete to distinguish between the two cases.
Let us define a relaxed $k$-edge colouring of $G$ as a function $f:E\to\{1,2,\dots,k\}$ such that for every vertex $v$ of $G$, at most two edges incident on $v$ have the same colour.
What is the complexity of relaxed edge colouring?
I am particularly interested in the complexity of relaxed $k$-edge colouring for graphs of maximum degree $k$.
Edit: The part below was part of the question earlier. But, it is wrong because proper $k$-edge colouring and thus relaxed $k$-edge colouring are trivially in P for graphs with maximum degree $k-1$.
I am particularly interested in the complexity of relaxed $k$-edge colouring for graphs of maximum degree $k-1$. I have a hunch the problem is going to be computationally hard, which is interesting because $k$-edge colouring is polynomial time solvable for the class.