As stated, the conjecture is false in an uninteresting way ... it is possible for a Cayley graph of Z_15 have a 5-cycle as a core.... So if we take M = Z_15 then the result is false.

Perhaps the question should either (a) be restricted to elementary abelian groups or (b) have the conclusion being that the core of a Cayley graph on M must be a Cayley graph on N where N is a (group) homomorphic image of M.

## Question needs refining...

As stated, the conjecture is false in an uninteresting way ... it is possible for a Cayley graph of Z_15 have a 5-cycle as a core.... So if we take M = Z_15 then the result is false.

Perhaps the question should either (a) be restricted to elementary abelian groups or (b) have the conclusion being that the core of a Cayley graph on M must be a Cayley graph on N where N is a (group) homomorphic image of M.

Gordon Royle http://people.csse.uwa.edu.au/gordon