Is it true that any finite graph has a $K\_n$ minor, where $n$ is a minimal vertex degree?



Does minimal degree $n$ imply a $K_n$ minor
*No.* 


The [edge-graph of the icosahedron](https://en.wikipedia.org/wiki/Regular_icosahedron#Icosahedral_graph) is regular of degree five, but does not have a $K\_5$ minor because it is planar ([Kuratowski's theorem](https://en.wikipedia.org/wiki/Kuratowski%27s_theorem)).


