1. **Problem Statement:** We are given 15 municipalities of Laguna represented as vertices numbered 1 to 15. We need to: - Draw edges between vertices if their municipalities are adjacent on the map. - Apply vertex coloring so no two adjacent vertices share the same color. - Determine the chromatic number (minimum colors needed). 2. **Step 1: Identify adjacency (edges) between vertices based on map borders.** From the map and municipality positions, the adjacency is: - 1 (Kalayaan) adjacent to 2 (Lumban) - 2 (Lumban) adjacent to 1, 3 (Santa Cruz), 5 (Cavinti) - 3 (Santa Cruz) adjacent to 2, 4 (Pagsanjan), 6 (Pila), 7 (Magdalena), 11 (Victoria) - 4 (Pagsanjan) adjacent to 3, 5 (Cavinti), 7 (Magdalena), 9 (Majayjay) - 5 (Cavinti) adjacent to 2, 4, 9 (Majayjay) - 6 (Pila) adjacent to 3, 7 (Magdalena), 10 (Liliw), 11 (Victoria) - 7 (Magdalena) adjacent to 3, 4, 6, 8 (Luisiana), 10 (Liliw), 11 (Victoria) - 8 (Luisiana) adjacent to 7, 9 (Majayjay), 10 (Liliw), 12 (Calauan) - 9 (Majayjay) adjacent to 4, 5, 8, 13 (Nagcarlan) - 10 (Liliw) adjacent to 6, 7, 8, 12 (Calauan), 14 (Rizal) - 11 (Victoria) adjacent to 3, 6, 7, 12 (Calauan), 15 (San Pablo) - 12 (Calauan) adjacent to 8, 10, 11, 13 (Nagcarlan), 14 (Rizal), 15 (San Pablo) - 13 (Nagcarlan) adjacent to 9, 12, 14 (Rizal) - 14 (Rizal) adjacent to 10, 12, 13, 15 (San Pablo) - 15 (San Pablo) adjacent to 11, 12, 14 3. **Step 2: Apply vertex coloring algorithm (Greedy Coloring):** - Assign color 1 to vertex 1. - Vertex 2 adjacent to 1, assign color 2. - Vertex 3 adjacent to 2, assign color 1 (different from 2). - Vertex 4 adjacent to 3 and 5 (not colored yet), assign color 2. - Vertex 5 adjacent to 2 and 4, assign color 1. - Vertex 6 adjacent to 3, assign color 2. - Vertex 7 adjacent to 3,4,6, assign color 3 (colors 1 and 2 taken). - Vertex 8 adjacent to 7, assign color 1. - Vertex 9 adjacent to 4,5,8, assign color 2. - Vertex 10 adjacent to 6,7,8, assign color 4 (colors 1,2,3 taken). - Vertex 11 adjacent to 3,6,7, assign color 4. - Vertex 12 adjacent to 8,10,11, assign color 3. - Vertex 13 adjacent to 9,12, assign color 1. - Vertex 14 adjacent to 10,12,13, assign color 2. - Vertex 15 adjacent to 11,12,14, assign color 1. 4. **Step 3: Determine chromatic number:** The highest color number used is 4. **Answer:** The chromatic number of the Laguna municipalities graph is **4**.