Countable graphs are majority 3choosable
Abstract
The Unfriendly Partition Conjecture posits that every countable graph admits a 2colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but it is known that a 3colouring with this property always exists. Anholcer, Bosek and Grytczuk recently gave a listcolouring version of this conjecture, and proved that such a colouring exists for lists of size 4. We improve their result to lists of size 3; the proof extends to directed acyclic graphs. We also discuss some generalisations.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.10408
 Bibcode:
 2020arXiv200310408H
 Keywords:

 Mathematics  Combinatorics;
 05C15 (Primary) 05C63;
 05C20 (Secondary)
 EPrint:
 6 pages. Minor changes including adding a reference