Abstrakt
Given eta in [0,1], a colouring C of V(G) is an eta-majority colouring if at most eta deg^+(v) out-neighbours of v have colour C(v), for any vertex v. We show that every digraph G equipped with an assignment of lists L, each of size at least k, has a 2/k-majority L-colouring.
For even k this is best possible, while for odd k the constant 2/k cannot be replaced by any number less than 2/(k+1). This generalizes a result of Anholcer, Bosek and Grytczuk, who proved the cases k = 3 and k = 4 and claim a weaker result for general k.