~y_{1} ~y_{2} ~y_{3} ~y_{4} ~y_{5} ~y_{6} ~y_{7} ~y_{8})^4$$ is nonzero, then there are values from $L$ for $x_1,\ldots,x_7,y_1,\ldots,y_8$ such that $f$ is nonzero by Theorem \[CN\]. Using the Magma algebra system [@magma], this monomial has coefficient $-2$, and thus there is a strong edge-coloring using the 5-list assignment[^1]. Thus, the 8-caterpillar does not exist in a vertex minimal counterexample.

Note that the proof in Lemma \[lma:caterpillar\] cannot be extended to exclude a 7-caterpillar in $G$, as there exists a 5-coloring of the external edges that does not extend to the caterpillar, even when the lists are all the same.

To complete the proof, we apply a discharging argument to $\operatorname{ct}(G)$. [^2]. First, observe that by Lemma \[lma:cutedge\], $\operatorname{ct}(G)$ is 2-connected and so every face is a simple cycle of length at least 41. Also observe that by Lemma \[lma:caterpillar\], $\operatorname{ct}(G)$ does not contain a path of length 8 where every vertex is of degree 2, unless one of those vertices is $p$.

Assign charge $2d(v)-6$ to every vertex $v \neq p$, charge $\ell(f) - 6$ to every face $f$, and charge $2d(p)+5$ to $p$. By Proposition \[prop:sum\], the total amount of charge on $\operatorname{ct}(G)$ is $-1$. Apply the following discharging 