Defective List Colorings of Planar Graphs

We combine the concepts of list colorings of graphs with the concept of defective colorings of graphs and introduce the concept of defective list colorings. We apply these concepts to vertex colorings of various classes of planar graphs. A defective coloring with defect d is a coloring of the vertices such that each color class corresponds to an induced subgraph with maximum degree at most d. A k-list assignment L, is an assignment of sets to the vertices so that |L(v)| = k, for all vertices v and an L-list coloring is a coloring such that the color assigned to v is in L(v) for all vertices v, and a d-defective L-list coloring is an L-list defective coloring with defect d. For a given graph G and defect d, we are interested in the smallest number k such that any k-list assignment, L, is d-defective L-list colorable. We will show that for outerplanar graphs, any 2-list assignment, L, has a 2-defective L-list coloring, and that this is best possible. We give results of this form pertaining to triangle-free outerplanar graphs and bipartite planar graphs. In general, we prove that all planar graphs are 2-defective L-list colorable for any 3-list assignment L.