1. **Problem Statement:** We need to find the chromatic number of the given graph using the Welch-Powell algorithm, which colors the graph with the minimum number of colors such that no two adjacent vertices share the same color. 2. **Welch-Powell Algorithm Overview:** - Step 1: Order vertices by descending degree (number of edges). - Step 2: Assign the first color to the highest degree vertex. - Step 3: Assign the same color to other vertices not adjacent to any vertex already colored with that color. - Step 4: Repeat with new colors until all vertices are colored. 3. **Calculate degrees of vertices:** - A: 2 (B, C) - B: 3 (A, D, E) - C: 3 (A, D, F) - D: 4 (B, C, E, F) - E: 5 (B, D, F, G, I, H) [Note: E connects to B, D, F, G, I, H] - F: 4 (C, D, E, G) - G: 4 (E, F, H, J) - H: 4 (E, G, I, J) - I: 3 (E, H, J) - J: 3 (G, H, I) Ordering vertices by degree descending: $$E(6), D(4), F(4), G(4), H(4), B(3), C(3), I(3), J(3), A(2)$$ 4. **Coloring process:** - Color 1: Assign to E. - Color 1: Cannot assign to D, F, G, H (adjacent to E). - Color 1: Can assign to B? B adjacent to E, no. - Color 1: B adjacent to E, so no. - Color 1: C? Adjacent to D and F, not E, but D and F uncolored yet, so safe to assign? But C adjacent to D and F, which are uncolored, so no conflict yet. But since D and F uncolored, we can assign color 1 to C if not adjacent to E. C not adjacent to E, so assign color 1 to C. - Color 1: A? Adjacent to B and C. C is color 1, so A cannot be color 1. - Color 2: Assign to D (next highest degree uncolored). - Color 2: Assign to F? Adjacent to D (color 2), so no. - Color 2: Assign to G? Adjacent to E (color 1) and F (uncolored), so safe. But G adjacent to E (color 1), no conflict with color 2, so assign color 2 to G. - Color 2: Assign to H? Adjacent to E (color 1) and G (color 2), so no. - Color 3: Assign to F. - Color 3: Assign to H? Adjacent to E (color 1), G (color 2), I (uncolored), J (uncolored), so safe. Assign color 3 to H. - Color 4: Assign to B. - Color 4: Assign to I? Adjacent to E (color 1), H (color 3), J (uncolored), so safe. Assign color 4 to I. - Color 5: Assign to J. - Color 6: Assign to A. 5. **Summary Table:** | Vertex | Degree | Color | |--------|--------|-------| | E | 6 | 1 | | C | 3 | 1 | | D | 4 | 2 | | G | 4 | 2 | | F | 4 | 3 | | H | 4 | 3 | | B | 3 | 4 | | I | 3 | 4 | | J | 3 | 5 | | A | 2 | 6 | 6. **Chromatic Number:** The minimum number of colors needed is 6. 7. **Graph Coloring Representation:** The original graph structure is maintained with vertices colored as above.