More connectedness of colorings
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?
Possibly relevant posts:
- Connectedness of the set of feasible -colorings (2/21/2007)
- Coloring Problem (12/21/2004)
- Planarity of a bipartite graph (10/1/2007)