Chromatic number χ(G): minimum colors for proper vertex coloring (adjacent vertices differ). χ(K_n)=n, χ(tree)=2, χ(C_{odd})=3, χ(planar)≤4 (Four Color Theorem). Computing χ is NP-hard, but bounds exist: ω(G)≤χ(G)≤Δ(G)+1.
How to Use
Select a graph:
- χ(G): Chromatic number
- Coloring: Example assignment
- Bounds: Clique and degree
Four Color Theorem
Every planar graph has χ≤4. Proved by Appel-Haken (1976) using computers — the first major theorem proved by computer. Verified independently by Robertson-Sanders-Seymour-Thomas (1997). Still no human-only proof exists!
Bounds
Lower: ω(G)≤χ(G) (clique number). Upper: χ(G)≤Δ(G)+1 (greedy). Brooks: χ≤Δ unless G is complete or odd cycle. Perfect graphs: χ=ω (Lovász, SPGT 2006).
Step-by-Step Instructions
- 1Select graph.
- 2Compute χ(G).
- 3View coloring.
- 4Check bounds.
- 5Compare to clique.