Graph coloring: assign colors to vertices so no adjacent vertices share a color. χ(G) = minimum colors needed. Four Color Theorem: χ(G) ≤ 4 for all planar graphs. P(G,k) = chromatic polynomial counts k-colorings. P(K_n,k) = k(k-1)(k-2)...(k-n+1). NP-hard in general.
How to Use
Define graph edges:
- χ(G): Minimum colors
- P(G,k): Chromatic polynomial
- Coloring: Valid assignment
Four Color Theorem
Every planar graph can be properly colored with ≤4 colors. Proven 1976 by Appel & Haken using computer-assisted proof (first of its kind). Still no human-only proof exists. 5 colors is easy to prove; 4 is the hard part.
Complexity
Finding χ(G) is NP-hard. Even deciding if χ(G)≤3 is NP-complete. Greedy coloring uses ≤Δ(G)+1 colors (Brook's theorem: ≤Δ unless complete or odd cycle). For planar: always ≤4 (Four Color Theorem).
Step-by-Step Instructions
- 1Define edges.
- 2Compute χ(G).
- 3Find coloring.
- 4Compute polynomial.
- 5Verify properness.