Planar graphs are 1-relaxed, 4-choosable

We show that every planar graph G=(V,E)G=(V,E) is 1-relaxed, 4-choosable. This means that, for every list assignment LL that assigns a set of at least four colors to each vertex, there exists a coloring ff such that f(v)∈L(v)f(v)∈L(v) for every vertex v∈Vv∈V and each color class f−1(α)f−1(α) of ff induces a subgraph with maximum degree at most 11.