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An automorphism $\alpha$ of a Cayley graph $Cay(G,S)$ of a group $G$ withconnection set $S$ is color-preserving if $\alpha(g,gs) = (h,hs)$ or$(h,hs^{-1})$ for every edge $(g,gs)\in E(Cay(G,S))$. If every color-preservingautomorphism of $Cay(G,S)$ is also affine, then $Cay(G,S)$ is a CCA (Cayleycolor automorphism) graph. If every Cayley graph $Cay(G,S)$ is a CCA graph,then $G$ is a CCA group. Hujdurovi\'c, Kutnar, D.W. Morris, and J. Morris haveshown that every non-CCA group $G$ contains a section isomorphic to thenonabelian group $F_{21}$ of order $21$. We first show that there is a uniquenon-CCA Cayley graph $\Gamma$ of $F_{21}$. We then show that if $Cay(G,S)$ is anon-CCA graph of a group $G$ of odd square-free order, then $G = H\timesF_{21}$ for some CCA group $H$, and $Cay(G,S) = Cay(G,T)\Box\Gamma$.

On Color Preserving Automorphisms of Cayley Graphs of Odd Square-free Order