Generalized Majority Colourings of Digraphs

Abstract

We almost completely solve a number of problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural numberk, the smallest numberm=m(k) such that every digraph can be coloured withmcolours where each vertex has the same colour as at most a 1/kproportion of its out-neighbours. We show thatm(k) ∈ {2k− 1,2k}. We also prove a result supporting the conjecture thatm(2) = 3. Moreover, we prove similar results for a more general concept called majority choosability.