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We show that any minimal Cayley graph of a (finitely generated) generalized dihedral or nilpotent group has chromatic number at most 3, while 4 colors are sometimes necessary for soluble groups. On the other hand we construct graphs of unbounded chromatic number that admit a proper edge coloring such that each cycle has some color at least twice. The latter disproves a conjecture of Babai'78 that would have implied that all minimal Cayley graphs have bounded chromatic number -- a problem that remains open.
García-Marco et al. (Wed,) studied this question.