Defective (improper) coloring: allow up to d monochromatic edges per vertex. χ_d(G): minimum colors for d-defective coloring. χ_0 = χ (proper). As d increases, fewer colors needed. Planar: χ_2 ≤ 3 (Lovász), χ_1 ≤ 4 (conjectured 3).
How to Use
Select graph and defect:
- d: Defect level
- χ_d: Defective χ
- vs χ: Compare
Planar Results
Planar, d=0: χ ≤ 4 (4CT). d=1: χ_1 ≤ 4, conjectured ≤ 3. d=2: χ_2 ≤ 3 (Lovász 1966, Cowen-Cowen-Woodall 1986). d≥5: χ_d ≤ 2 (trivial high defect). Beautiful tradeoff between colors and defects.
Applications
Wireless frequency assignment: small interference tolerable (defect). Scheduling: limited conflicts acceptable. Graph partitioning: bounded-degree subgraphs. Practical relaxation of strict coloring constraints.
Step-by-Step Instructions
- 1Select graph.
- 2Choose defect d.
- 3Compute χ_d.
- 4Compare with χ_0=χ.
- 5Analyze tradeoff.