$(\le\!5)$-cycles, and the subgraph $G[f]$ of $G$ drawn in the closure of $f$ contains no triangles. By Theorem \[thm-53\], every precoloring of the cycle bounding $f$ extends to a $3$-coloring of $G[f]$. Consequently, a precoloring of the rings of $G$ extends to a $3$-coloring of $G$ if and only if it extends to a $3$-coloring of $H$.

In Section \[sec-htw\], we describe possible colorings of the graphs appearing in Theorems \[thm-many\] and \[thm-crtri\]. Section \[sec-bc\] is devoted to examining a chain $G$ of graphs, many of which are not quadrangulations, showing that either all precolorings of the rings of $G$ extend, or that the rings of $G$ have length $4$ and all their precolorings which extend to a $3$-coloring of the reduced Thomas-Walls graph $T'_4$ also extend to $G$. This allows us to prove Theorem \[thm-crtri\] in Section \[sec-crtri\]. In Section \[sec-qua\], we continue by examining chains that contain a long subchain consisting only of quadrangulations and show that either every precoloring of the rings extends or the chain is a near $3,3$-quadrangulation. We combine these results and prove Theorem \[thm-many\] in Section \[sec-many\].

Colorings of the special graphs {#sec-htw}
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In this section we study which precolorings of rings 