Theorem \[thm-6cyc\], there exists a $3$-coloring $\varphi_1$ of the patch replacing $u'_2$ such that $\varphi_1(x)=\psi(x)$ for $x\in\{u_1,u_2,u_3\}$ and $\varphi_1(z)=\varphi'(z)$, where $z$ is the neighbor of $u'_2$ distinct from $u_1$ and $u_3$. Let $\varphi_2$ be defined analogously at $C_2$. Observe that $\psi\cup\varphi_1\cup \varphi_2\cup (\varphi'\restriction V(G))$ is a $3$-coloring of $G$ extending $\psi$.

Using this lemma, we can easily handle Havel-Thomas-Walls graphs as well.

\[lemma-col-htw\] Let $G$ be a graph obtained by framing on the interface pair $v_1v_3$ of a patched Havel-Thomas-Walls graph, with ring $C=v_1v_2v_3v_4$. A $3$-coloring $\psi$ of $C$ extends to a $3$-coloring of $G$ if and only if $\psi$ is not dangerous on $C$.

Consider first the case that $G$ is a Havel-Thomas-Walls graph, obtained from the reduced Thomas-Walls graph $T'_n$ with rings $C$ and $C'=u_1u_2u_3u_4$ and interface pairs $v_1v_3$ and $u_1u_3$ by either adding the edge $u_1u_3$, or the graph in Figure \[fig-havel\](a). Consider any $3$-coloring $\varphi$ of $G$. Note that both the edge $u_1u_3$ and the graph from Figure \[fig-havel\](a) ensure that $\varphi(u_1)\neq\varphi(u_3)$, and thus $\varphi$ is $u_1$-diagonal on $C'$. Consequently, $\varphi$ is dangerous on $C'$, and by Lemma \[lemma-col-tw\], it is not dangerous on $C$. Therefore, if $\psi$ extends to a $3$-coloring of $G$, then $\psi$ 