Majority Colourings of Digraphs
  We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3. 
Stephan Kreutzer and Sang-il Oum and Paul Seymour and Dominic van der   Zypen and David R. Wood
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