$WS_m$, assume that $m \neq 5$ and consider the same assignment as before plus $f(w) = 1$. Then, values of $f(N(v))$ remains the same as in $CS_m$, values of $f(N(u))$ are the same as in $CS_m$ plus one, $f(N_{WS_m}(u)) = f(N_{CS_m}(u))+1$, and $f(N(w)) = \lceil 3m/2 \rceil$. If $m = 4$, clearly $f$ is an additive 2-coloring of $WS_4$. If $m \geq 6$, $f(N(w)) > 8 \geq f(N(u))$ and $f$ is an additive 2-coloring of $WS_m$.

For $m = 5$, we propose a different additive 2-coloring of $WS_5$: $f(u_1) = f(u_2) = f(u_4) = f(v_4) = f(v_5) = 1$, $f(u_3) = f(u_5) = f(v_1) = f(v_2) = f(v_3) = f(w) = 2$. Then, $f(N(v_1)) = 2$, $f(N(v_2)) = 3$, $f(N(u_5)) = 6$, $f(N(w)) = 7$, $f(N(u_1)) = f(N(u_3)) = 8$, $f(N(u_2)) = f(N(u_4)) = 9$.

Let $m \geq 3$. Then, $\eta(KS_m) = \bigl\lceil \frac{m+2}{3} \bigr\rceil.$

For the sake of simplicity, we call $r = \lceil \frac{m+2}{3} \rceil$. Note that $d(u_i) = m+1$ for all $i$. In virtue of Corollary \[COTAINF\], we have $\eta(G) \geq r$.

We only have to propose an additive $r$-coloring of $KS_m$. First, define a permutation function $\mathfrak{p}:[m] \rightarrow [m]$ as follows: $\mathfrak{p}(1) = 1$, $\mathfrak{p}(j) = 