Prove that a k-regular connected graph with edge chromatic number = k is 2-connected. Prove that a $k$-regular connected graph with edge chromatic number = $k$ is $2$-connected.
I think from handshaking lemma that k must be odd but i don't see how this helps.
any help much appreciated

Hint:


*

*If you would pick one particular color, edges of that color would form a perfect matching.

*Union of any two disjoint perfect matchings forms a set of disjoint simple cycles spanning the whole graph.

*Assume there is a cut vertex $u$, and its removal would form at least $2$ connected components, pick some two colors $\alpha$ and $\beta$ with edges leading to different components.

*Union of matchings implied by $\alpha$ and $\beta$ forms cycles, one of which passes through $u$, hence, $u$ couldn't have been a cut vertex.


I hope this helps $\ddot\smile$
