Let $G$ be a critical graph drawn in the sphere with holes.

-   Every vertex $v\in V(G)$ that does not belong to the rings has degree at least three.

-   If $K$ is a $(\le\!5)$-cycle in $G$ forming the boundary of an open disk $\Lambda$, and the closure of $\Lambda$ does not contain any contractible triangle of $G$, then $\Lambda$ is a face of $G$.

-   If $K$ is a closed walk of length $6$ in $G$ forming the boundary of an open disk $\Lambda$, and the closure of $\Lambda$ does not contain any contractible triangle of $G$, then either $\Lambda$ is a face of $G$, or all faces of $G$ contained in $\Lambda$ have length $4$.

Let $C$ be the union of the rings of $G$, and consider any vertex $v\in V(G)\setminus V(C)$. Suppose for a contradiction that $v$ has degree at most $2$. Let $\psi$ be any $3$-coloring of $C$ that extends to a $3$-coloring $\varphi$ of $G-v$. Then $\psi$ also extends to a $3$-coloring of $G$, by giving the vertices of $V(G)\setminus\{v\}$ the same color as in the coloring $\varphi$ and by choosing a color of $v$ distinct from the colors of its neighbors. This contradicts the 