TITLE: Bound on chromatic numbers of union of graphs
QUESTION [6 upvotes]: 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!

REPLY [2 votes]: The edges of the complete graph on $n$ vertices can be covered using roughly $\log n$ bipartite graphs; so there exists a graph $G$ and bipartite graph $H$ such that adding $H$ to $G$ increases the chromatic number by $\dfrac{n}{\log n}$.