make use of the following, which is another simple and well known application of Euler’s Formula.

\[prop:sum\] In a planar graph $G$,

$$\sum_{f \in F(G)} ( \ell(f) - 6 ) + \sum_{v \in V(G)} (2d(v) - 6) = -12.$$

We will also use the Combinatorial Nullstellensatz, which will be applied to show we can extend certain list colorings.

\[CN\] Let $f$ be a polynomial of degree $t$ in $m$ variables over a field $\mathbb{F}$. If there is a monomial $\prod x_i^{t_i}$ in $f$ with $\sum t_i=t$ whose coefficient is nonzero in $\mathbb{F}$, then $f$ is nonzero at some point of $\prod S_i$, where each $S_i$ is a set of $t_i+1$ distinct values in $\mathbb{F}$.

The first item of Theorem \[thm:mainlist\] follows from the following strengthened theorem.

Let $G$ be a planar $\{1,3\}$-regular graph of girth at least $41$, and let $p \in V(G)$. Assign distinct colors to the edges incident to $p$ and let $L$ be a $5$-list-assignment to the remaining edges of $G$. There exists a strong edge-coloring $c$ where $c(e) \in L(e)$ for all $e \in E(G)$.

For the sake of contradiction, select $G$, $p$, $c$, and $L$ as in the theorem statement, and assume there does not 