is *weak*. A $3$-coloring $\psi$ of $C$ is *dangerous* if either $\psi$ is $v_1$-diagonal, or $C$ is strong and $\psi$ is bichromatic.

Let us first deal with Thomas-Walls graphs.

\[lemma-col-tw\] Let $n\ge 1$ be an integer, let $H$ be a patched Thomas-Walls graph obtained from $T'_n$ by patching, and let $G$ be a graph obtained by framing on interface pairs $u_1u_3$ and $v_1v_3$ of $H$. Let $C_1=u_1u_2u_3u_4$ and $C_2=v_1v_2v_3v_4$ be the rings of $G$ and let $\psi$ be a precoloring of $C_1\cup C_2$. If $\psi$ extends to a $3$-coloring of $G$, then it is dangerous on at most one of $C_1$ and $C_2$. Furthermore, if $n\ge 4$ and $\psi$ is not dangerous on both $C_1$ and $C_2$, then $\psi$ extends to a $3$-coloring of $G$.

Firstly, suppose that $G$ is the reduced Thomas-Walls graph $T'_n$. We proceed by induction on $n$. The claims obviously hold when $n=1$. Hence, assume that $n\ge 2$, and thus both $C_1$ and $C_2$ are strong. Let $G$ be obtained from a copy $G'$ of $T'_{n-1}$ with rings $C_1$ and $C'_2=v'_1v'_2v'_3v_4$ (with interface pairs $u_1u_3$ and $v'_2v_4$) by adding the ring $C_2$ and the edge $v_2v'_2$. Let $\psi$ be a $3$-coloring of $C_1\cup C_2$.

Suppose for 