On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism (alpha ) of a Cayley graph (mathrm{Cay}(G,S)) of a group G with connection set S is color-preserving if (alpha (g,gs) = (h,hs)) or ((h,hs^{-1})) for every edge ((g,gs)in E(mathrm{Cay}(G,S))). If every color-preserving automorphism of (mathrm{Cay}(G,S)) is also affine, then (mathrm{Cay}(G,S)) is a Cayley color automorphism (CCA) graph. If every Cayley graph (mathrm{Cay}(G,S)) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group (F_{21}) of order 21. We first show that there is a unique non-CCA Cayley graph (Gamma ) of (F_{21}). We then show that if (mathrm{Cay}(G,S)) is a non-CCA graph of a group G of odd square-free order, then (G = Htimes F_{21}) for some CCA group H, and (mathrm{Cay}(G,S) = mathrm{Cay}(H,T)mathbin {square }Gamma ).