A note on list improper coloring of plane graphs

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