Star chromatic number χ_s(G): minimum proper coloring where every 2-colored subgraph is a star forest (collection of stars). Equivalently: no path P₄ is 2-colored. Strictly stronger than acyclic coloring. χ_s(planar) ≤ 20.
How to Use
Select graph:
- χ_s: Star χ
- Check: No 2-color P₄
- vs a: Compare acyclic
Coloring Hierarchy
χ ≤ a ≤ χ_s. Proper ⊂ acyclic ⊂ star. Each level adds restrictions. Star coloring: every 2-colored component is a star (not just a forest). Most restrictive of the standard variants.
Upper Bounds
Planar: χ_s ≤ 20 (Albertson et al). Bounded degree Δ: χ_s = O(Δ^(3/2)). Trees: χ_s = ⌈(Δ+1)/2⌉ + 1 or similar. Many open problems remain for specific graph families.
Step-by-Step Instructions
- 1Select graph.
- 2Compute χ_s.
- 3Check P₄ condition.
- 4Compare with a(G).
- 5Apply bounds.