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 φ that assigns a color φ(v) ∈ L(v) to each vertex v ∈ V (G) such that at most d neighbors of v receive color φ(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. © 2008 Elsevier B.V. All rights reserved.