HMMT 2009 T6

For any set of graphs $G_{1}, G_{2}, \ldots, G_{n}$ all having the same set of vertices $V$, define their overlap, denoted $G_{1} \cup G_{2} \cup \cdots \cup G_{n}$, to be the graph having vertex set $V$ for which two vertices are adjacent in the overlap if and only if they are adjacent in at least one of the graphs $G_{i}$. (a) $[\mathbf{1 0}]$ Let $G$ and $H$ be graphs having the same vertex set and let $a$ be the chromatic number of $G$ and $b$ the chromatic number of $H$. Find, in terms of $a$ and $b$, the largest possible chromatic number of $G \cup H$. Prove your answer.