1. Problem: Determine the chromatic number of the graph G = (V, E) where V is the set of cities in Metro Manila and E is the set of edges representing adjacency between cities. 2. The Four-Color Theorem states that any planar graph can be colored with at most four colors such that no two adjacent vertices share the same color. 3. Since the graph G represents cities and their adjacencies on a map (which is planar), the chromatic number $\chi(G)$ satisfies: $$\chi(G) \leq 4$$ 4. Therefore, the minimum number of colors needed to color the vertices of G so that no two adjacent cities share the same color is 4. This completes the solution for the first problem.