For each k > 1 construct a k-regular simple graph having no 1-factor.
How would I do this?
If k is even consider the clique with k+1 vertices, if k is odd then do the following:
1) In a k+1 clique where the vertices are , delete the edges .
2) Insert a new vertex and join it to .
3) Do this to k different such cliques and create corresponding .
4) Introduce a vertex v" and join it to .
Can you see why this graph will not have a perfect matching ?