Planar graphs are 1-relaxed, 4-choosable

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