copies of $K_n$ which share a single vertex, $W_n^m = m K_{n-1} \lor K_1$. Then, $\eta(W_n^m) = n-1$.

Let $n$ be an integer such that $n \geq 4$. A *wheel* is defined as $W_n = C_n \lor K_1$, where $C_n$ is a circuit of $n$ vertices. In order to know $\eta(W_n)$ we first need to know $\eta(C_n)$. Although there already exists a manuscript written by Akbari, Assadi, Emamjomeh-Zadeh and Khani giving the additive chromatic number of circuits, here we propose a different and short proof of it for the sake of completeness.

Let $n \geq 4$. If $n$ is even, then $\eta(C_n) = 2$. Otherwise, $\eta(C_n) = 3$.

If $n$ is even, $C_n$ is a regular bipartite graph and we can use Corollary \[REGULARBIP\]. So, we prove that $\eta(C_n) = 3$ for $n$ odd. Let $V = \{v_1, \ldots, v_n\}$ and suppose that $f:V \rightarrow \{1,2\}$ is an additive 2-coloring of $C_n$. Then, $f$ is also a topological additive 2-numbering of a certain digraph $D$ such that $G(D) = C_n$. Observe that $f(N(v)) \in \{2,3,4\}$ for all $v \in V$. Since $C_n$ is not bipartite, there must be an oriented path of 3 consecutive vertices in $D$. W.l.o.g. assume that 