at least $k$ in $\operatorname{ct}(G)$.

We will make use of the discharging method for some of our results. For an introduction to this method, see the survey by Cranston and West [@CW]. We will directly use two standard results that can be proven using this method. Both of Theorems \[thm:bi\] and \[thm:wz\] rely on Lemmas \[lma:cw\] and \[lma:nrs\].

Let $G$ be a graph and $\operatorname{ct}(G$) be the contracted graph. An *$\ell$-thread* is a path $v_1\dots v_\ell$ in $\operatorname{ct}(G)$ where each $v_i$ is a $2{{\ensuremath{^\perp}}}$-vertex.

\[lma:cw\] If $G$ is a graph with girth at least $\ell+1$ and $\operatorname{mad}(G) < 2 + \frac{2}{3\ell - 1}$, then $\operatorname{ct}(G)$ contains a 1-vertex or an $\ell$-thread.

\[lma:nrs\] If $G$ is a planar graph with girth at least $5\ell + 1$, then $\operatorname{ct}(G)$ contains a 1-vertex or an $\ell$-thread.

Strong List Edge-Coloring of Subcubic Graphs {#sec:list}
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In this section, we prove Theorem \[thm:mainlist\]. Our proof uses the discharging method, wherein we assign an initial charge to the vertices and faces of a theoretical minimal counterexample. This initial charge is then disbursed according to a set of discharging rules in order to draw a contradiction to the existence of such a minimal counterexample. We will often 