the edge $(u_i, v_i)$, note first that $Q \subsetneqq V \backslash \{v_i\}$. Hence, $f(N(v_i)) = f(Q) - f(u_i) < f(V \backslash \{v_i\}) - f(u_i) = f(N(u_i))$.\
We finish by proving that $\eta(G) \geq r$. Suppose that there exists an additive $(r-1)$-coloring $f$ of $G$. Recall that $f(N(u_i)) = f(V) - f(u_i) - f(v_i)$ for all $i \in [q]$. Thus, $f(V) - (2r-2) \leq f(N(u_i)) \leq f(V) - 2$. Since there are $2r-3$ integers in the range of feasible values for $f(N(u_i))$ and $2r-3 < q$, there are two indexes $j$ and $k$ such that $f(N(u_j))=f(N(u_k))$ by the pigeonhole principle, leading to a contradiction.

Suns
====

Let $G$ be a graph and $U = \{u_1, \ldots, u_m\} \subset V(G)$. A *sun* is a graph $G'$ with $V(G') = V(G) \cup V$ where $V = \{v_1, \ldots, v_m\}$ and $$E(G') = E(G) \cup \{(u_i, v_{i-1}), (u_i, v_i) ~:~ i \in [m] \}.$$ For the sake of simplicity, $u_0$ and $v_0$ are another names for vertices $u_m$ and $v_m$.

In this section, we study *cycle suns* $CS_m$, when $G$ is a circuit ($V(G) = U$ and $E(G) = \{ (u_i, u_{i-1}) ~:~ i \in [m] \}$), *wheel suns* $WS_m$, when $G$ is a 