List colourings of planar graphs

Let G = (V,E) be a graph, let f : V (G)→ N, and let k ≥ 0 be an integer. A list-assignment L of G is a function that assigns to each vertex v of G a set (list) L(v) of colors: usually each color is a positive integer. We say that L is an f -assignment if |L(v)| = f(v) for all v ∈ V , and a k-assignment if |L(v)| = k for all v ∈ V . A coloring ofG is a function φ that assigns a color to each vertex ofG so that φ(v) 6= φ(w) whenever vw ∈ E. An L-coloring of G is a coloring φ of G such that φ(v) ∈ L(v) for all v ∈ V . If G admits an L-coloring, then G is L-colorable. The graph G is said to be f list-colorable and f is called a choice function if G is L-colorable for every f -assignment L of G. When f(v) = k for all v ∈ V , the corresponding term becomes k-list-colorable or k-choosable. The list-chromatic number or choice number χ ` (G) is the least number k such that G is k-list-colorable. In 1979 Erdős, Rubin and Taylor [1] conjectured that all planar graphs are 5-list-colorable, but that not all planar graphs are 4-list-colorable. Both conjectures were proved in 1993.