for every $v\in V(G_0)$. Suppose that $\varphi_1\cup \psi$ is not a $3$-coloring of $G'$. By Lemma \[lemma-cylflow\], the winding numbers of $\varphi_1$ on $K_1$ and $K_2$ are the same, and thus $\varphi_1(x_{2,j})=\psi(v_{2,j})$ for $j\in\{1,2,3\}$. Then, it follows that $\psi\cup\varphi_2$ is a $3$-coloring of $G'$. We conclude that $G'$, and thus also $G$, has a $3$-coloring which extends $\psi$.

Let $G$ be a near $3,3$-quadrangulation, let $C$ be one of the rings of $G$ and let $\psi$ be a $3$-coloring of $C$. Let us discuss several cases:

-   If $C$ shares an edge with a triangle $T$ in $G$ (where possibly $T=C$ if $C$ is a triangle), then $\psi$ uniquely extends to a $3$-coloring $\varphi$ of $T$. Let $w$ be the winding number of $\varphi$ on $T$. In this case, we say that $\psi$ on $C$ *causes winding number $w$*.

-   If $C$ does not share an edge with a triangle, then let $C=v_1v_2v_3v_4$, where $v_1v_2v_3$ is a part of the boundary of a $5$-face $f$ (note that the choice of $v_2$ is unique). If $\psi(v_1)\neq \psi(v_3)$, then draw an edge between $v_1$ and $v_3$ in $f$, and let $w$ be the winding number of $\psi$ on the triangle $v_1v_3v_4$. In 