\dots, c_6$ are the colors of the edges incident to the vertices $v_0$ and $v_9$, and assign variables $y_1, \dots, y_8$ to the pendant edges, and variables $x_1, \dots, x_7$ to the interior edges as shown in Figure \[fig:lcat\].

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

in [1,2,...,8]{}[ (,0)–(,1) (,0) node\[free\] (,1) node\[extra\]; ]{};

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

in [1,2,...,8]{}[ (-.2,.5) node[$y_{\x}$]{}; ]{}

in [1,2,...,7]{}[ (+.5, -.2) node[$x_{\x}$]{}; ]{}

(.5,-.2) node[$c_{3}$]{} (8.5,-.2) node[$c_{4}$]{} (-.25,.6) node[$c_1$]{} (-.25,-.6) node[$c_2$]{} (9.25,.6) node[$c_5$]{} (9.25,-.6) node[$c_6$]{} ;

(-1,-.75) node\[insep\]; (-1,.75) node\[insep\]; (10,-.75) node\[insep\]; (10,.75) node\[insep\];

Identifying the conflicts between variables and colors produces the following polynomial, $$\begin{aligned}
f(y_1, \dots, y_8,x_1,\dots, x_7) &= (y_2 - c_3)(x_2 - c_3)(y_7 - c_4)(x_6 - c_4) \\
&\quad\cdot 
\prod_{i=1}^3(x_1 - c_i)\prod_{i=1}^3(y_1 - c_i) \prod_{i=4}^6 (x_7 - c_i) \prod_{i=4}^6 (y_8 - c_i)\\ 
&\quad\cdot \prod_{j -i  \in \{1,2\}}(x_i-x_j)  \prod_{j-i = 1}(y_i - y_j)\prod_{i - j \in \{-1,0,1,2\}} (y_i - x_j). \end{aligned}$$

We will use the Combinatorial Nullstellensatz to show that there is an assignment of colors $\hat{c}_1,\dots, \hat{c}_8$ and $c_1',\dots, c_7'$ such that $f(\hat{c}_1,\dots,\hat{c}_8,c_1',\dots,c_7')\ne 0$. Such an assignment of colors would extend the inductive coloring of $G-p$ to a strong edge-coloring of $G$. If the coefficient of $$(x_{1}~ x_{2}~ x_{3}~ x_{4} ~x_{5} ~x_{6} ~x_{7} 