Majority choosability of countable graphs

In any vertex coloring of a graph some edges have differently colored ends (bichromatic edges) and some are monochromatic. In a proper coloring all edges are bichromatic. In a majority coloring it is enough that for every vertex v, the number of monochromatic edges incident to v does not exceed the number of bichromatic edges incident to v. A well known result proved by Lovasz asserts that every finite graph has a majority 2coloring. A similar statement for countably infinite graphs is a challenging open problem, known as the Unfriendly Partition Conjecture. We consider a natural list variant of majority coloring. A graph is majority k-choosable if it has a majority coloring from any lists of size k assigned arbitrarily to the vertices. We prove that every countable graph is majority 4-choosable. We also consider a natural analog of majority coloring for directed graphs. We prove that every countable digraph is also majority 4choosable. We pose list and directed analogs of the Unfriendly Partition Conjecture, stating that every countable graph is majority 2-choosable and every countable digraph is majority 3choosable. (c) 2023 Elsevier Ltd. All rights reserved.