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

Abstract

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 K r ( T ) denote the set of r-kings of T. Let Q be the subdigraph of T induced by K 4 ( 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 K 4 ( Q ) ⊆ K 4 ( T ) , we further prove that (3) K 3 ( Q ) ⊆ K 3 ( T ) and (4) K 2 ( Q ) = K 2 ( 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.