A note on list improper coloring planar graphs

A graph G is called (k, d)*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ϵ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4-cycles and l-cycles for some l ϵ {5, 6, 7} is (3, 1)*-choosable.