A note on list improper coloring of plane graphs

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v∈V(G). An (L,d)^∗-coloring is a mapping ? that assigns a color ?(v)∈L(v) to each vertex v∈V(G) such that at most d neighbors of v receive color ?(v). A graph is called (k,d)^∗-choosable, if G admits an (L,d)^∗-coloring for every list assignment L with |L(v)|≥k for all v∈V(G). In this note, it is proved that every plane graph, which contains no 4-cycles and l-cycles for some l∈{8,9}, is (3,1)^∗-choosable.