strong edge-colorable, discovered by computer. Therefore, $\operatorname{ct}(G)$ is not isomorphic to a cycle, and hence for every $2{{\ensuremath{^\perp}}}$-vertex $u$ in $G$, $|N_3(u)| \geq 1$.

If $G$ has some vertex $v$ such that $|N_3(v)|=1$, then $G$ must be a subgraph of $\Theta(t_1,t_2,t_3)$, which is the graph consisting of three internally disjoint $x-y$ paths of length $t_1+1, t_2+1$ and $t_3+1$, for some $0 \leq t_1 \leq t_2 \leq t_3$.

If $t_3 \geq 8$, then $\operatorname{ex}_3(G)$ contains an 8-caterpillar, so we assume that $t_3 < 8$. Observe that if $\operatorname{mad}(\Theta(t_1,t_2,t_3)) < 2 + \frac{1}{7}$, then $t_1+t_2+t_3 \geq 13$. However, if $\Theta(t_1,t_2,t_3)$ does not contain a reducible $Y$-type configuration, then by Claim \[clm:Reducible\] the sequence $(t_1,t_2,t_3)$ is one of $(0,7,7)$, $(0,6,7)$, $(1,6,6)$, $(1,5,7)$, $(2,5,6)$, $(2,4,7)$, or $(3,3,4)$. In each of these cases, we have verified by computer that $\Theta(t_1,t_2,t_3)$ has a strong edge-coloring using five colors.

Therefore, $|N_3(v)|\geq 2$ for every $v\in\operatorname{ct}(G)$. We proceed using discharging. Assign each vertex initial charge $d(v)$. Note that the total charge on the graph is $2e(\operatorname{ct}(G))$, which is at most $\operatorname{mad}(G)n < (2+\frac{1}{7})n$. We shall distribute charge among the vertices of $\operatorname{ct}(G)$ and result with charge at least $2 + \frac{1}{7}$ on every vertex, giving a contradiction.

Distribute 