Equitable chromatic number χ_=(G): minimum k for proper k-coloring where each color class has ⌊n/k⌋ or ⌈n/k⌉ vertices. Hajnal-Szemerédi (1970): χ_=(G) ≤ Δ(G)+1. Beautiful scheduling application. Chen-Lih-Wu conjecture: χ_= ≤ Δ for most graphs.
How to Use
Select graph:
- χ_=: Equitable χ
- Balance: Class sizes
- H-S: ≤ Δ+1
Hajnal-Szemerédi Theorem
Every graph G has equitable (Δ+1)-coloring. Proof by Kierstead-Kostochka (2008) gives polynomial algorithm. Strengthens Brooks' theorem in the equitable setting. One of the most important results in graph coloring.
Scheduling Applications
Equitable coloring models fair task scheduling: each time slot (color) gets roughly equal workload. Exam scheduling: distribute exams evenly across time slots. Load balancing: equal partition among processors.
Step-by-Step Instructions
- 1Select graph.
- 2Compute χ_=.
- 3Check class balance.
- 4Apply H-S bound.
- 5Use for scheduling.