assumption that $G$ is critical.

The other two conclusions of the lemma follow by a similar argument using Theorem \[thm-6cyc\].

Furthermore, colorings of tents can be described using Theorem \[thm-6cyc\]. Consider a $4$-cycle $C=u_1u_2u_3u_4$ and its $3$-coloring $\psi$. We say that $\psi$ is *$u_1$-diagonal* if $\psi(u_1)\neq \psi(u_3)$, and that it is *bichromatic* if $\psi(u_1)=\psi(u_3)$ and $\psi(u_2)=\psi(u_4)$. Note that every $3$-coloring of $C$ is $u_1$-diagonal, $u_2$-diagonal, or bichromatic.

\[cor-col-pt\] If $G$ is a tent with the ring $C=v_1v_2v_3v_4$, then exactly the $v_1$-diagonal and $v_2$-diagonal colorings of $C$ extend to $3$-colorings of $G$.

Let $G_0$ be either a reduced Thomas-Walls graph, or a Havel-Thomas-Walls graph. Let $S$ be an independent set of vertices of $G_0$ of degree three. Let $C_0=v_1v_2v_3v_4$ be a ring of $G_0$, with the interface pair $v_1v_3$. Let $G_1$ be obtained from $G_0$ by replacing the vertices of $S$ by patches, and let $G$ be obtained from $G_1$ by framing on its interface pairs. Let $C=v_1v'_2v_3v'_4$ be the ring of $G$ corresponding to $C_0$. We say that the ring $C$ is *strong* if $G\neq C$, $v_2,v_4\not\in S$, $v'_2=v_2$, and $v'_4=v_4$, that is, $v_2$ and $v_4$ are not affected by patching or created by framing. Otherwise, we say that $C$ 