the ones bounded by $C$ and $T$ have length $4$. Furthermore, if $\psi$ is the $3$-coloring of $C$ given by $\psi(v_1)=\psi(v_3)=1$, $\psi(v_2)=\psi(v_4)=2$ and $\psi(v_5)=3$ and $\psi$ does not extend to a $3$-coloring of $G$, then $v_2$ and $v_3$ are incident with $T$.

As a part of the proof of Theorem \[thm-many\], we need to prove a strengthening Theorem \[thm-onetri\] and describe the $3$-coloring properties of graphs embedded in the disk with a ring $C$ of size $4$ and with exactly two triangles. All such critical graphs are described in Theorem \[thm-crtri\] below, which is of independent interest.

To state this result, let us give another construction related to Thomas-Walls graphs. For $n\ge 1$, let $G=T'_n$ be a reduced Thomas-Walls graph with rings $C_1=u_1u_2u_3u_4$ and $C_2=v_1v_2v_3v_4$ and interface pairs $u_1u_3$ and $v_1v_3$. Consider the embedding of $T'_n$ in the disk, obtained by closing the hole in the face bounded by $C_1$. Let $G'$ be a graph obtained from $G$ by either adding the edge $u_1u_3$, or the subgraph depicted in Figure \[fig-havel\](a) (this graph is often called *Havel’s quasiedge*, since Havel [@conj-havel] used it to disprove a conjecture by Grünbaum that every planar graph without intersecting triangles is 3-colorable). A *patched 