exist a strong edge coloring of $E(G)$ using colors from $L$. In this selection, minimize $n(G)$. Note that $G$ is connected and $e(G) > 5$. We can further assume that $d(p) > 1$, since if $d(p)=1$ and $\{p'\}=N(p)$ then we can instead color the edges incident to $p'$.

\[lma:cutedge\] There does not exist a cut-edge $uv$ such that $d(u) = d(v) = 3$.

Suppose that $G$ contains a cut-edge $uv$ with $d(u) = d(v) = 3$. There are exactly two components in $G - uv$, call them $G_1$ and $G_2$, with $u \in V(G_1)$ and $v \in V(G_2)$. Without loss of generality, $p \in V(G_1)$. For each $i \in \{1,2\}$, let $G_i' = G_i + uv$.

Since $d(v) = 3$, $n(G_1') < n(G)$. Thus there is a strong edge-coloring of $G_1'$ using the 5-list-assignment $L$. Next, color the other two edges incident to $v$ using colors distinct from those on the edges incident to $u$. Now, $G_2'$ is a subcubic planar graph of girth at least 41 with distinctly colored edges about the vertex $v$ and $n(G_2') < n(G)$. Thus, there is an extension of the coloring to $G_2'$.

The colorings of $G_1'$ and $G_2'$ form a strong edge coloring 