Grundy number Γ(G): maximum colors any greedy first-fit coloring can use. For the WORST vertex ordering, how many colors does greedy need? χ(G) ≤ Γ(G) ≤ Δ(G)+1. A graph is well-colored if Γ = χ (no bad orderings).
How to Use
Select graph:
- Γ: Grundy number
- Best: χ colors
- Worst: Γ colors
Well-Colored Graphs
G is well-colored if Γ(G) = χ(G): no ordering makes greedy use extra colors. Complete graphs, cycles, trees are well-colored. Not all graphs are: some have orderings forcing many extra colors.
Computing Γ
NP-hard to compute (Zaker, 2006). Even determining Γ=k is coNP-complete. For trees: polynomial (Γ(T) is height of the tree in a specific rooting). For P_n: Γ=⌈log₂ n⌉+1.
Step-by-Step Instructions
- 1Select graph.
- 2Find Γ(G).
- 3Find worst order.
- 4Compare to χ.
- 5Check well-colored.