Consider a symmetric or arc-transitive graph except the odd cycle. Then, is it true that the graph could be partitioned into distinct parts such that each part has equal number of vertices except for a few singleton parts(which consists of single vertex) and such that each vertex in each part is adjacent to some other vertex in every other part except the part in which it is(the vertices in a part are independent) and the number of parts is $\le\Delta+1$ where $\Delta$ is the maximum degree?
I think this should be true, as any the graph is both vertex and edge transitive, hence any edge as well as vertex must be equivalent under an automorphism, thus the partitions must also be equivalent. Is the fact true for graphs that are only vertex transitive and not necessarily symmetric or edge transitive? What about Cayley graphs? or, restrictively, Cayley graphs of abelian groups? Thanks beforehand.