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For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. 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.
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However, the maximum list chromatic number of planar graphs is 5, not 4, so the extension fails already for -minor-free graphs. [10] More generally, for every t ≥ 1 {\displaystyle t\geq 1} , there exist graphs whose Hadwiger number is 3 t + 1 {\displaystyle 3t+1} and whose list chromatic number is 4 t + 1 {\displaystyle 4t+1} .