Local structure of 5 and 6-connected graphs

We prove, that if graph on $n$ vertices is mimimally and contraction critically 5-connected, then it has $4n/7$ vertices of degree 5. We also prove, that if graph on $n$ vertices is mimimally and contraction critically 6-connected, then it has $n/2$ vertices of degree 6. Bibl. 7 titles.