Prove that $\lambda (G) \leq r/2$ if $\kappa(G) = 1$ of r-regular connected graph $G$ and $r > 1$

Question

This is a question I found on exercise book. But I dont think it is true.

For example, this graph:

It is an upsided triangle on the top, and a triangle at the bottom. In this case, $\lambda(G) > r/2$.

Answer 1

That graph isn't regular. The middle vertex has $4$ neighbors, but the other vertices have $2$.

Answer 2

Same idea, if $κ(G)=1$ it has a cut vertex, now use Menger's theorem which states that you will have $λ(G)$ edgedisjoint paths between two nodes of the graph, now if our nodes are on either side of the cutvertex then the maximum amount of such paths is clearly $\frac r2$