Choosability ch(G): minimum k for list coloring from ANY k-lists. ch ≥ χ always. Can be strictly greater: K_{3,3} has χ=2 but ch=3. Thomassen (1994): ch(planar)=5. Galvin (1995): ch(L(K_{n,n}))=n (list edge coloring conjecture for bipartite).
How to Use
Select graph:
- ch: Choosability
- Gap: ch vs χ
- Lists: Worst-case assignment
Thomassen's Theorem
Every planar graph is 5-choosable (Thomassen, 1994). Elegant induction proof. Not 4-choosable in general (Voigt, 1993 counterexample). The list coloring conjecture: ch' = χ' for all graphs (open!).
List Coloring Conjecture
ch'(G) = χ'(G)? (List edge chromatic = edge chromatic.) Proven for bipartite graphs (Galvin, 1995). Open in general. Would imply every graph's edges are list-colorable with Δ+1 colors from any Δ+1-lists.
Step-by-Step Instructions
- 1Select graph.
- 2Compute ch(G).
- 3Compare to χ.
- 4Find worst lists.
- 5Apply theorems.