Grundy number Γ(G): maximum colors any greedy (first-fit) coloring can use. Depends on vertex ordering. χ(G) ≤ Γ(G) ≤ Δ(G)+1. Measures how badly greedy can perform. NP-hard to compute. Important for algorithm analysis.
How to Use
Select graph:
- Γ: Grundy number
- vs χ: Compare
- Greedy: Worst case
Greedy Coloring
Greedy: process vertices in order, assign smallest available color. Best ordering: uses χ colors. Worst ordering: uses Γ colors. Gap can be huge: χ=2 but Γ=n/2 for some bipartite graphs!
Bounds
χ ≤ Γ ≤ Δ+1. For trees: Γ = Δ+1 (tight!). For bipartite: Γ can be up to ⌈n/2⌉. Grundy number is hard to compute (NP-hard) and hard to approximate. Active research area.
Step-by-Step Instructions
- 1Select graph.
- 2Try orderings.
- 3Apply greedy.
- 4Find worst case.
- 5Compare Γ vs χ.