Properties
Choosability ch(G) satisfies the following properties for a graph G with n vertices, chromatic number χ(G), and maximum degree Δ(G):
- ch(G) ≥ χ(G). A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
- ch(G) cannot be bounded in terms of chromatic number in general, that is, ch(G) ≤ f(χ(G)) does not hold in general for any function f. In particular, as the complete bipartite graph examples show, there exist graphs with χ(G) = 2 but with ch(G) arbitrarily large.
- ch(G) ≤ χ(G) ln(n).
- ch(G) ≤ Δ(G) + 1.
- ch(G) ≤ 5 if G is a planar graph.
- ch(G) ≤ 3 if G is a bipartite planar graph.
Read more about this topic: List Coloring
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