List Colorings of K5-Minor-Free Graphs With Special List Assignments

A list assignment L of G is a function that assigns to every vertex v of G a set (list) L(v) of colors. The graph G is called L-list colorable if there is a coloring φ of the vertices of G such that φ(v) ∈ L(v) for all v ∈ V (G) and φ(v) 6= φ(w) for all vw ∈ E(G). We consider the following question of Bruce Richter, where d(v) denotes the degree of v in G: Let G be a planar, 3-connected graph that is not a complete graph. Is G L-list colorable for every list assignment L with |L(v)| = min{d(v), 6} for all v ∈ V ? This is joint work with Anja Pruchnewski, Zsolt Tuza, and Margit Voigt.