to be disjoint. Let $C$ be the union of the rings of such a graph $G$ ($C$ is empty when $G$ is embedded in the sphere without holes). By a *precoloring* of $C$, we mean any proper $3$-coloring of $C$. We say that $G$ is *critical* if $G\neq C$ and for every proper subgraph $G'$ of $G$ such that $C\subseteq G'$, there exists a precoloring of $C$ that extends to a $3$-coloring of $G'$, but not to a $3$-coloring of $G$; that is, removing any edge or vertex not belonging to $C$ affects the set of precolorings of $C$ that extend to a $3$-coloring of the graph.

We construct a sequence of graphs $T_1$, $T_2$, …, which we call *Thomas-Walls graphs* (Thomas and Walls [@tw-klein] proved that they are exactly the $4$-critical graphs that can be drawn in the Klein bottle without contractible cycles of length at most $4$). Let $T_1$ be equal to $K_4$. For $n\ge 1$, let $u_1u_3$ be any edge of $T_n$ that belongs to two triangles and let $T_{n+1}$ be obtained from $T_n-u_1u_3$ by adding vertices $x$, $y$ and $z$ and edges $u_1x$, $u_3y$, $u_3z$, $xy$, $xz$, and $yz$. The first few graphs of this sequence 