Improper choosability and Property B

Abstract

A fundamental connection between list vertex colorings of graphs and Property B (also known as hypergraph 2-colorability)was already known to Erd˝os, Rubin, and Taylor ((1980), 125–157). In this article, we draw similar connections for improper list colorings. This extends results of Kostochka (Electron J Combin 9 (2002)), Alon (Combin Probab Comput 1 (1992), 107–114; (1993), 1–33; Random Structures Algorithms 16 (2000), 364–368), and Kr´ al’ and Sgall (J Graph Theory 49 (2005), 177–186) for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.