We denote vertices of $Q$ with $u_1,\ldots,u_q$ and vertices of $S$ with $v_1,\ldots,v_s$. W.l.o.g. we assume that $Q$ is maximal (unless stated otherwise). The following result states an upper bound of the additive chromatic number of split graphs.

\[COTASUPSPLIT\] Let $G = (Q \cup S, E)$ be a split graph where $Q$ is maximal and $T \subset Q$ be a non-empty set such that the degrees of each vertex of $T$ differ each other. Then, $\eta(G) \leq |Q|-|T|+1$.

W.l.o.g. let $T = \{u_{q-t+1},u_{q-t+2},\ldots,u_{q-1}, u_q\}$ where $t = |T|$. We exhibit an additive $(q-t+1)$-coloring of $G$. Consider the assignment $f:V \rightarrow [q-t+1]$ such that $f(u_i) = i$ for all $i \in [q-t]$, $f(w) = q-t+1$ for all $w \in T \cup S$. We first check for edges between the clique and the stable set. Let $(u_i, v) \in E$. Since $Q$ is maximal, for each $v \in S$, there exists $u(v) \in Q$ such that $v$ is not adjacent to $u(v)$. Then, $f(N(v)) \leq f(Q) - f(u(v)) \leq f(Q) - 1$. On the other hand, let $r_i = |N(u_i) \cap S|$ for all $i \in [q]$. Since $v \in N(u_i)$, $r_i \geq 1$ and $f(N(u_i)) = f(Q) - f(u_i) + (q-t+1).r_i 