of degree 2, except when using $p$, so, $r_2 \leq \left\lfloor \frac{7}{8}\ell(f)\right\rfloor$. Since $\ell(f) \geq 41$, we have $$\ell(f) - r_2 - \frac{3}{2} \geq \ell(f) - \left\lfloor \frac{7}{8}\ell(f)\right\rfloor - \frac{3}{2} \geq 0.$$ Thus, all vertices and faces have nonnegative charge, contradicting Proposition \[prop:sum\].

The second item of Theorem \[thm:mainlist\] follows by similarly strengthening the statement to include a precolored vertex and using Lemmas \[lma:cw\], \[lma:cutedge\], and \[lma:caterpillar\].

Strong Edge-Coloring of Sparse Graphs {#sec:sparse}
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In this section, we prove Theorems \[thm:sparse\] and \[thm:sparse4\].

Let $G$ be a graph with maximum degree $\Delta(G) \leq d$. For a vertex $v$ in $\operatorname{ct}(G)$ denote by $N_3(v)$ the set of $3^+$-vertices $u$ where $\operatorname{ct}(G)$ contains a path $P$ from $u$ to $v$ where all internal vertices of $P$ are $2{{\ensuremath{^\perp}}}$-vertices. For $u \in N_3(v)$, let $\mu(v,u)$ be the number of paths from $v$ to $u$ whose internal vertices have degree 2 in $\operatorname{ct}(G)$. For a 3-vertex $v$, let the *responsibility set*, denoted $\operatorname{Resp}(v)$, be the set of $2{{\ensuremath{^\perp}}}$-vertices that appear on the paths between $v$ and the vertices in $N_3(v)$.

Let $D$ be a subgraph of $G$. We call $D$ a *$k$-reducible configuration* if there exists a subgraph $D'$ of $D$ such that any 