a closed walk tracing $C$ in the positive direction. We define the *winding number* of $\psi$ on $C$ as $\omega(W,\psi)/3$. Observe that the winding number is an integer of the same parity as the length of $C$. We need the following result concerning the colorability of quadrangulations, see e.g. the Propositions 4.1 and 4.2 in [@trfree5].

\[lemma-cylflow\] Let $G$ be a graph embedded in the cylinder with rings $C_1$ and $C_2$, such that all other faces of $G$ have length $4$. For any $3$-coloring $\varphi$ of $G$, the winding number of $\varphi$ on $C_1$ is equal to the winding number of $\varphi$ on $C_2$.

We also need a strengthening of Lemma \[lemma-cylflow\] for quadrangulations of cylinder with rings of length at most $4$.

\[lemma-col44\] Let $G$ be a tame graph embedded in the cylinder with rings $C_1$ and $C_2$ of length at most $4$. If all faces of $G$ have length $4$, the distance between $C_1$ and $C_2$ is at least $|C_1|$, and $\psi$ is a precoloring of $C_1\cup C_2$ that does not extend to a $3$-coloring of $G$, then $|C_1|=|C_2|=3$ and $\psi$ has opposite winding numbers on $C_1$ and $C_2$.

Note that $C_1$ and $C_2$ have the same parity, 