I don't understand why if a graph contains two subgraphs which share one common vertex, the chromatic number of the graph is the maximum of that of the two.

I understand that the chromatic number for the entire graph should beat leastthe larger chromatic number.

But since the common vertex in the entire graph has more adjacent nodes than in the subgraph, is it not possible that we need more colors to avoid adjacent nodes having the same color?

In other words, how can we be sure that that at the degree of the common endpoint is less than or equal to the degree/chromatic number of the subraph with the larger chromatic number?