Graph polynomials and paintability of plane graphs

There exists a variety of coloring problems for plane graphs, involving vertices, edges, and faces in all possible combinations. For instance, in the entire coloring of a plane graph we are to color these three sets so that any pair of adjacent or incident elements get different colors. We study here some problems of this type from algebraic perspective, focusing on the facial variant. We obtain several results concerning the Alon–Tarsi number of various graphs derived from plane embeddings. This allows for extensions of some previous results for choosability of these graphs to the game theoretic variant, known as paintability. For instance, we prove that every plane graph is facially entirely 8-paintable, which means (metaphorically) that even a color-blind person can facially color the entire graph from lists of size 8.