ChapterZero

Ok, so I had a thought on how to maybe add edges to the coloring graph mentioned in the previous post so that it would become connected. Namely, in addition to the previous neighbor relation ( if the colorings differ in exactly one vertex), we could say if the partitionings of the graph corresponding to the two colorings are the same, but they assign a different color to each partition. At least this would eliminate the previous counterexample: the graph of 4 colorings on a fully connected 4 vertex graph is connected in this new sense. Does this hold for general graphs and general -colorings?