Acyclic chromatic number a(G): minimum colors for proper coloring where every pair of color classes induces a forest (acyclic). Stronger than proper coloring. Borodin (1979): planar → a ≤ 5. Grünbaum conjecture proved! Applications in Hessian computation.
How to Use
Select graph:
- a: Acyclic χ
- Check: No 2-color cycle
- vs χ: Compare
Borodin's Theorem
Every planar graph has a(G) ≤ 5. Proves Grünbaum's 1973 conjecture. The proof is intricate, using discharging. Tight: some planar graphs need a = 5. One of the jewels of graph coloring theory.
Hessian Computation
Computing sparse Hessian matrices: acyclic coloring determines the number of matrix-vector products needed. a(G) colors → Hessian computable in a(G) gradient evaluations. Important for optimization, machine learning.
Step-by-Step Instructions
- 1Select graph.
- 2Compute a(G).
- 3Check bichromatic cycles.
- 4Apply Borodin.
- 5Use for Hessian.