# What is the complexity of a special multigraph edge coloring problem

Given a multigraph such that there are 0 or 2 edges connecting every two vertices, we are to color the edges of this graph so that adjacent edges receive distinct colors. It is known that we need at least $$\Delta(G)$$ colors, and $$\Delta(G)+2$$ colors are sufficient, so its chromatic index is $$\Delta(G)$$, or $$\Delta(G)+1$$, or $$\Delta(G)+2$$. What is the complexity of determining the chromatic index of such a graph $$G$$? I guess it might be NP-Complete but is there any source mentioning this result? Thanks in advence!

I strongly suspect that this is NP-complete, but the approach I have in mind does not seem to work! I wanted to use the the well-known fact that it is NP-complete to decide whether the chromatic index $$\chi'(G)$$ of a (simple) cubic graph $$G$$ is $$3$$ or $$4$$ (see this paper of Holyer).
For the reduction, start with a simple cubic graph $$G$$ and double each edge to obtain a multigraph $$2G$$.
False Claim. $$\chi'(2G)=6$$ if and only if $$\chi'(G)=3$$.
To refute the false claim, let $$P$$ be the Petersen graph. It is well-known that $$\chi'(P)=4$$, but it is easy to check that $$\chi'(2P)=6$$.
If the false claim were true, then a polynomial-time algorithm to compute $$\chi'(2G)$$ would yield a polynomial-time algorithm to compute $$\chi'(G)$$.