5-choosability of Graphs with Crossings Far Apart

We give a new proof of the fact that every planar graph is 5choosable, and use it to show that every graph drawn in the plane so that the distance between every pair of crossings is at least 15 is 5-choosable. At the same time we may allow some vertices to have lists of size four only, as long as they are far apart and far from the crossings. Thomassen [5] gave a strikingly beautiful proof that every planar graph is 5-choosable. To show this claim, he proved the following more general statement: Theorem 1. Let G be a plane graph with the outer face F , xy an edge of F , and L a list assignment such that |L(v)| ≥ 5 for v ∈ V (G)\V (F ), |L(v)| ≥ 3 for v ∈ V (F ) \ {x, y}, |L(x)| = |L(y)| = 1 and L(x) 6= L(y). Then G is