Suppose, we know that $G$ is a regular graph of odd order that is $k$- edge choosable, where $k$ is the degree. Then, is it true that $\overline{G}$ has list edge chromatic number at most $n-k+1$?
I think the answer is yes. This is because, by Haggkvist-Janssen theorem, we have that the complete graph of odd order has list chromatic index $n$, where $n$ is the order. Thus, if edge choosability of $G$ is $k$, then it is straightforward to see that $\overline{G}$ has list edge chromatic number at most $n-k+1$ by a greedy approach. If this be true, we need to obtain only the list chromatic indices of graphs with $\Delta\ge\frac{n}{2}$, where $\Delta$ is the maximum degree. Am I right, or are there counterexamples? Thanks beforehand.
