extend in the special graphs appearing in Theorems \[thm-many\] and \[thm-crtri\]. We examine patched Havel-Thomas-Walls graphs in Lemma \[lemma-col-htw\] and tame almost $3,3$-quadrangulations in Lemma \[lemma-col-33\].

We need the following result of Gimbel and Thomassen [@gimbel].

\[thm-6cyc\] Let $G$ be a triangle-free graph drawn in the disk with a ring $C$ of length at most $6$. If $G$ is critical, then $|C|=6$ and every other face of $G$ has length $4$. Furthermore, a precoloring $\psi$ of $C=v_1\ldots v_6$ does not extend to a $3$-coloring of such a graph $G$ if and only if one of the following conditions holds.

-   There exists $i\in\{1,2,3\}$ such that $v_iv_{i+3}\in E(G)$ and $\psi(v_i)=\psi(v_{i+3})$, or

-   $\psi(v_1)=\psi(v_4)\neq\psi(v_2)=\psi(v_5)\neq\psi(v_3)=\psi(v_6)\neq \psi(v_1)$.

Theorem \[thm-6cyc\] has an important consequence for patches.

\[cor-patch\] Let $G$ be a patch with ring $xaybzc$, and let $H$ be the graph with vertex set $\{x,y,z,v\}$ and edges $xv$, $yv$ and $zv$. Then a $3$-coloring of $\{x,y,z\}$ extends to a $3$-coloring of $G$ if and only if it extends to a $3$-coloring of $H$.

That is, replacing a vertex of degree three by a patch does not affect the $3$-colorability of the graph. Let us give an observation about critical graphs that is often useful.

\[lemma-fr\] 