Greedy coloring: process vertices in order, assign smallest available color. Uses at most Δ+1 colors. Brooks (1941): at most Δ colors unless K_n or odd cycle. Welsh-Powell: order by decreasing degree for better results.
How to Use
Select graph and ordering:
- Greedy: Sequential assign
- Colors: At most Δ+1
- Order: Affects quality
Brooks' Theorem
For connected graphs: χ(G) ≤ Δ(G), except K_n and odd cycles. Greedy with right ordering achieves this. Brooks' theorem is tight: cubic graphs can need 3 colors (Petersen).
Vertex Orderings
Random: unpredictable quality. Largest-first (Welsh-Powell): sort by decreasing degree. Smallest-last (Matula): iteratively remove min-degree vertex. DSatur: choose vertex with most colored neighbors. Each gives different guarantees.
Step-by-Step Instructions
- 1Select graph.
- 2Choose ordering.
- 3Run greedy.
- 4Count colors.
- 5Compare to χ.