$C'_i$.

Suppose first for a contradiction that $\psi$ is dangerous both on $C_1$ and $C_2$, and that it extends to a $3$-coloring $\varphi$ of $G$.

-   If $C_1$ is weak, then $\psi$ is $u_1$-diagonal on $C_1$, and $\psi'(u'_i)=\varphi(u'_i)=\psi(u_i)$ for $i\in\{2,4\}$. Let $z$ be the neighbor of $u'_2$ distinct from $u_1$ and $u_3$. If $u'_2\not\in S$, then $u'_2z$ is an edge of $G$, and $\varphi(z)\neq\psi'(u'_2)$. If $u'_2\in S$, then $\varphi(z)\neq\psi'(u'_2)$ by Theorem \[thm-6cyc\] applied to the patch $P$ replacing $u'_2$ and the $3$-coloring of $P$ given by $\varphi.$

-   If $C_1$ is strong, then $u'_2=u_2$, $u'_4=u_4$, and $u'_2z$ is an edge of $G$.

Together with a similar argument applied at $C_2$, we conclude that $\psi'\cup(\varphi\restriction V(G_1))$ is a $3$-coloring of $G_1$ extending $\psi'$, which is a contradiction since $\psi'$ is dangerous both on $C'_1$ and $C'_2$ and the conclusions of Lemma \[lemma-col-tw\] are satisfied by $G_1$ as we argued before.

Next, suppose that $\psi$ is not dangerous on one of the rings, say on $C_1$, and that $n\ge 4$. Then $\psi'$ is not dangerous on $C'_1$, and thus it extends to a $3$-coloring $\varphi'$ of $G_1$. If $u'_2\not\in S$, then let $\varphi_1$ be an empty coloring. If $u'_2\in S$, then by 