where $|Q|=|S|=q$ and the set of edges between $Q$ and $S$ is $\{(u_i, v_i) ~:~ i \in [q] \}$. A *thick headless spider* of orden $q \geq 2$ is a split graph where $|Q|=|S|=q$ and the set of edges between $Q$ and $S$ is $\{(u_i, v_j) ~:~ i,j \in [q],~ i \neq j \}$. Equivalently, a thick headless spider is the complement of a thin headless spider of the same order and vice-versa.

Let $G$ be a thin/thick headless spider of order $q$. Then, $$\eta(G) = \biggl\lceil \dfrac{q+1}{2} \biggr\rceil.$$

For the sake of simplicity, we call $r = \lceil \frac{q+1}{2} \rceil$. We start by proving $\eta(G) = r$ when $G$ is thin. Note that $d(u_i) = q$ for all $i$. In virtue of Corollary \[COTAINF\], we have $\eta(G) \geq r$. Then, we only need to propose an additive $r$-coloring of $G$. If $q = 2$, consider the additive 2-coloring $f$ such that $f(u_1) = f(u_2) = f(v_1) = 1$ and $f(v_2) = 2$. If $q \geq 3$, consider the assignment $f:V \rightarrow [r]$ such that $f(u_i) = r - i + 1$ and $f(v_i) = 1$ for all $i \in [r]$, and $f(u_i) = q - i + 1$ and 