Here's the statement to be proved:
The chromatic number of a graph is the same as the maximum chromatic number of its blocks.
Here's what I think. We consider the graph $G$ with chromatic number $\chi(G)=k$. Let $\chi(B)$ be the maximum chromatic number of a block of $G$. Now, clearly $\chi(B) \leq k,$ since a block is a subgraph of $G.$ This would tell me that every individual block could be colored with at most $k$ colors.
Now, any two blocks share at most one vertex, so intuitively it would make sense that we could find a proper coloration for $G$ with at most $k$ colors, by picking the color of the common vertices in a clever way. But how can I prove this?
Any help is appreciated.