of $G$, a contradiction.

Define a *$k$-caterpillar* to be a $k$-thread $v_1,\dots,v_k$ in $G$ where $p \notin \{v_1,\dots,v_k\}$. Figure \[8cat\] is an $8$-caterpillar.

(0,0)–(9,0) (0,0) node\[insep\] (9,0) node\[insep\];

in [1,2,...,8]{}[ (,0)–(,1) (,0) node\[free\] (,1) node\[extra\]; at (,-0.35) [$v_{\x}$]{}; at (,1.5) [$v_{\x}'$]{}; ]{};

(9,0)–(10,.75) (9,0)–(10,-.75) (0,0)–(-1,.75) (0,0)–(-1,-.75);

(-1,-.75) node\[insep\]; (-1,.75) node\[insep\]; (10,-.75) node\[insep\]; (10,.75) node\[insep\]; at (-0.5,0) [$v_{0}$]{}; at (9.5,0) [$v_{9}$]{}; at (10.35,.75) [$v_9'$]{}; at (10.35,-.75) [$u_9'$]{}; at (-1.35,.75) [$v_0'$]{}; at (-1.35,-.75) [$u_0'$]{};

\[lma:caterpillar\] $G$ does not contain an $8$-caterpillar.

We will show that if $G-p$ contains an 8-caterpillar, then $G$ has a strong edge $L$-coloring. If $v_1,\dots,v_8$ form an 8-caterpillar, then let $v_i'$ be the 1-vertex adjacent to $v_i$, $v_0$ and $v_9$ be the other neighbors of $v_1$ and $v_8$. For $i \in \{0,9\}$, let $v_i'$ and $u_i'$ be the neighbors of $v_i$ other than $v_1$ or $v_8$. By removing all edges incident to $v_2,\dots, v_7$ and $u_1,\dots,u_8$, as well as any isolated vertices that are produced, we obtain a graph $G'$ with fewer vertices than $G$, so we can strongly edge-color $G'$ with 5 colors. We fix such a coloring of $G'$ and generate a contradiction by extending this coloring to a strong edge-coloring of $G$. Suppose that $c_1, 