If $X$ is any set, we let $[X]^2:=\big\{\{x,y\}:x\neq y\in X\big\}$. If $G=(V,E)$ is an undirected graph and $v\in V$, we define $N_G(v) = \{w\in V:\{v,w\}\in E\}$.

For $i =1,2$, let $G_i=(V_i,E_i)$ be non-empty, finite, simple, undirected graphs with $V_1\cap V_2 = \emptyset$. Suppose that $E \subseteq [V_1\cup V_2]^2$ has $E\cap [V_i]^2=E_i$ for $i=1,2$, and in addition has the following property:

there is $k\in\mathbb{N}$ such that $|N_G(v_1) \cap V_2| = k$ for all $v_1\in V_1,$

where $G := (V_1\cup V_2, E)$.

Is there a "global constant" $c_k\in\mathbb{N}$ depending on $k$ only such that we have $$\chi(G) \leq \max\{\chi(G_1), \chi(G_2)\} + c_k,$$ no matter what the initial choice of $G_1, G_2$ and $E$ (adhering to the property above) was?