Alternating kernels

An arc-coloured digraph D is alternating if whenever (u,v) and (v,w) are arcs in D they are of different colours. In an arc coloured digraph D, a subset K⊂V(D) of vertices of D is a kernel by alternating paths (or alternating kernel) if it is absorbent and independent by directed alternating paths. In this paper we prove sufficient conditions for the existence of alternating kernels in arc-coloured tournaments, quasi-transitive digraphs, and k-partite digraphs.