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 and , either has a vertex adjacent to all vertices in or has a vertex adjacent to all vertices in .
We show that a graph that has an over-dominating colouring with colours has a complete minor of order at least and that this bound is essentially best possible.