On the kings and kings-of-kings in semicomplete multipartite digraphs

Koh and Tan showed in (Evaluation of the number of kings in a multipartite tournament, submitted for publication.) that the subdigraph induced by the 4-kings of an n-partite tournament with no transmitters, where n⩾3, contains no transmitters. We extend this result to the class of semicomplete n-partite digraph, where n⩾2. Let T be a semicomplete multipartite digraph with no transmitters and let Kr(T) denote the set of r-kings of T. Let Q be the subdigraph of T induced by K4(T). In this paper, we (1) show that Q has no transmitters, (2) obtain some results on the 2-kings, 3-kings and 4-kings in T. While it is trivial that K4(Q)⊆K4(T), we further prove that (3) K3(Q)⊆K3(T) and (4) K2(Q)=K2(T). Maurer (Math. Mag. 53 (1980) 67) introduced the concept of kings-of-kings in tournaments. Following Maurer, we investigate the r-kings-of-kings of semicomplete multipartite digraphs with no transmitters for r=1,2,3,4. Some problems on the r-kings-of-kings are posed.