# Existence of perfect matching with low weight

Let $$k\ge1$$, and suppose that $$G$$ is a $$k$$-regular and $$(k−1)$$-edge-connected graph with an even number of vertices, and with edge weights $$c:E(G)\to\mathbb{R}$$.

Question: Is there always a perfect matching $$M$$ in $$G$$ with $$c(M)\le \frac1k\cdot c(E(G))$$.

Edit: Below are my original thoughts on this problem which turned out to be wrong.

It would obviously suffice to show that there are $$k$$ pairwise-disjoint perfect matchings of $$G$$. Let's try to prove this by induction:

If $$k=2$$ then $$G$$ is a circle with an even number of vertices and the claim easily follows.

Now assume that $$G$$ is $$(k+1)$$-regular and $$k$$-edge-connected. It easily follows from Tutte's Theorem that $$G$$ has a pefect matching $$M$$. Now it would be tempting to apply the inductive hypothesis to $$G-M$$. However while it is obvios that $$G-M$$ is $$k$$-regular it need not be $$(k-1)$$-edge-connected.

Here is an example of a $$3$$-regular, $$2$$-edge-connected graph $$G$$ with perfect matching (red) $$M$$ s.t. $$G-M$$ is not connected (i.e. $$1$$-edge-connected):

But maybe we just picked the wrong matching. We could have also choosen the following one which works:

So it would suffice if every $$(k+1)$$-regular and $$k$$-edge-connected graph $$G$$ with an even number of vertices contained a perfect matching $$M$$ s.t. $$G-M$$ is $$(k-1)$$-edge-connected? (Unfortunetly this turned out to be wrong as shown in the answers)