Wikipedia says
A k-regular graph is 1-factorable if it has chromatic index k.
If I am correct, it means that if a k-regular graph has chromatic index k, then it is 1-factorable.
Although it doesn't imply this, is it true that if a k-regular graph is 1-factorable, then it has chromatic index k? Thanks.
Let $G$ be a $k$-regular graph that is $1$-factorable. Consider a $1$-factorization of $G$. Since $G$ is $k$-regular the $1$-factorization is an edge coloring of $G$ that uses $k$ colors. Thus $\chi'(G)=k$.