is not dangerous on $C$.

Conversely, if $\psi$ is not dangerous on $C$, then we can extend it to a $3$-coloring of $T'_n$ that is $u_1$-diagonal on $C'$ (for $n\ge 4$, this follows by Lemma \[lemma-col-tw\]; for $1\le n\le 3$, it is easy to construct the desired colorings), and further extend the coloring to the graph from Figure \[fig-havel\](a) if present in $G$.

The case that $G$ is obtained from a Havel-Thomas-Walls graph by patching and framing is handled in the same way as in the proof of Lemma \[lemma-col-tw\].

Next, we consider the colorability of near $3,3$-quadrangulations. We need some additional definitions. Given a $3$-coloring $\psi:V(G)\to\{0,1,2\}$ of a graph $G$, let us define an orientation $G_\psi$ of $G$ by orienting every edge $uv\in E(G)$ towards $v$ if and only if $\psi(v)-\psi(u)\in\{1,-2\}$. Let $W=v_1v_2\ldots v_k$ be a walk in a graph $G$. We define $\omega(W,\psi)$ to be the difference between the number of forward and backward edges of $W$ in $G_\psi$, i.e., $$\omega(W,\psi)=|\{i:1\le i\le k-1,v_iv_{i+1}\in E(G_\psi)\}|-|\{i:1\le i\le k-1,v_{i+1}v_i\in E(G_\psi)\}|.$$ Suppose that $G$ is embedded in the cylinder, and let $C$ be a non-contractible cycle in $G$. We fix one orientation around the cylinder as *positive*. Let $W$ (with $v_1=v_k$) be 