there are graphs $G$ such that $\eta(G \lor K_q) = \max \{ \eta(G), q \}$ for any $q$. An example is the family of stable graphs. In that case, $G \lor K_q$ is called complete split. In the next section, we prove that the additive chromatic number of complete splits is $q$.

![A counterexample for $q = 2$.](W63.eps)

\[CONTRAEJEMPLO\]

The theorem also shows that if the conjecture holds for a graph $G$ then it still holds for $G \lor K_q$ (with $q \leq n-\Delta-1$) since $\chi(G \lor K_q) = \chi(G) + q$.\
A vertex $v$ is *universal* in a graph $G$ when $N(v) = V(G) \backslash \{v\}$. We will use a simplified version of Theorem \[UNIVERTICES\] for solving ACP on known families of graphs having a single universal vertex:

If $G$ is a graph without universal vertices, $\eta(G \lor K_1) = \eta(G)$.

Let $n$ be an integer such that $n \geq 3$. A $n$-*fan* is defined as $F_n = P_{n+1} \lor K_1$ where $P_{n+1}$ is a path of length $n$. Since $\eta(P_{n+1}) = 2$ (see [@ORLOW]), $\eta(F_n) = 2$.

Let $n, m$ be integers such that $n \geq 3$, $m \geq 2$. The *windmill* graph $W_n^m$ is defined as $m$ 