that the following holds. Let $G$ be a tame graph embedded in the cylinder with rings $C_1$ and $C_2$ of length at most $4$. If $G$ is a chain of at least $c$ graphs, then

-   every precoloring of $C_1\cup C_2$ extends to a $3$-coloring of $G$, or

-   $G$ contains a subgraph $H$ obtained from a patched Thomas-Walls graph by framing on its interface pairs, with rings $C_1$ and $C_2$, or

-   $G$ contains a near $3,3$-quadrangulation $H$ with rings $C_1$ and $C_2$ as a subgraph.

Aksenov [@aksenov] proved the following strengthening of Grötzsch’s theorem (fixing a previous flawed proof of this fact by Gr[ü]{}nbaum [@grunbaum]).

\[thm-3t\] Every planar graph with at most $3$ triangles is $3$-colorable.

In the course of the proof, Aksenov also established two interesting facts.

\[thm-onetri\] Let $G$ be a graph drawn in the cylinder with a ring $C$ of length at most $4$. If every triangle in $G$ is non-contractible, then every precoloring of $C$ extends to a $3$-coloring of $G$.

\[thm-53\] Let $G$ be a graph embedded in the disk with a ring $C=v_1v_2v_3v_4v_5$. Suppose that $G$ contains exactly one triangle $T$. If $G$ is critical, then all faces of $G$ other than 