Hadwiger numbers and over-dominating colourings

Motivated by Hadwiger’s conjecture, we say that a colouring of a graph is over-dominating if every vertex is joined to a vertex of each other colour and if, for each pair of colour classes C 1 and C 2 , either C 1 has a vertex adjacent to all vertices in C 2 or C 2 has a vertex adjacent to all vertices in C 1 . w that a graph that has an over-dominating colouring with k colours has a complete minor of order at least 2 k / 3 and that this bound is essentially best possible.