The paper mentioned above has been published:
Stephan Kreutzer, Sang-il Oum, Paul Seymour, Dominic van der Zypen, David R. Wood. "Majority Colourings of Digraphs" Electronic Journal of Combinatorics, 24(2):#P2.25, 2017. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p25
Other papers have appeared on this topic:
Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk. "Every digraph is majority 4-choosable" http://arxiv.org/abs/1608.06912
Fiachra Knox, Robert Šámal. "Linear Bound for Majority Colourings of Digraphs" https://arxiv.org/abs/1701.05715
A. Girao, T. Kittipassorn, and K. Popielarz. "Generalised majority colourings of digraphs" https://arxiv.org/abs/1701.03780