Some graphs ($K_n$ or $K_n$ minus any one edge, for instance) only permit one minimal colouring up to different labels of the colours. Is there anything known about these kind of graphs? I can think of a number of examples of such graphs (mostly just $K_{n,m}$ minus an edge or two), but I can't think of any general statements I can make about them.

I was just wondering if there was any prior research into this as I couldn't find anything. It could be that it's trivial (in which case I'd be interested in whether there's anything known about the number of minimal colourings).