Bound on chromatic numbers of union of graphs If I have a vertex-set $V$ and two graphs $G, H$ on $V$, it is easy to show that the chromatic number $\chi (G \cup H) \leq \chi (G) \chi (H)$. My question now is, whether $\chi (G \cup H) \leq \chi (G) + \chi (H)$ is also true; I guess this is not the case, but I couldn't find any counterexample. Does someone know one? Thanks!

Let $V$ be the vertices of a hexagon. Let $G$ be the graph that connects each node to its neighbors and second-nearest neighbors. Then $\chi(G)=3$, since the vertices can be colored "RGBRGB", and the inclusion of triangles means that $G$ is not $2$-colorable. Let $H$ be the graph connecting each node to its diametric opposite (i.e., three disjoint edges). Clearly $\chi(H)=2$. But $G\cup H=K_6$, so $$\chi(G\cup H)=6 > 5=\chi(G)+\chi(H).$$