On semicomplete multipartite digraphs whose king sets are semicomplete digraphs

Abstract

Reid [Every vertex a king, Discrete Math. 38 (1982) 93–98] showed that a non-trivial tournament H is contained in a tournament whose 2-kings are exactly the vertices of H if and only if H contains no transmitter. 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 ) . Very recently, Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249–258] proved that Q contains no transmitters and gave an example to show that the direct extension of Reid's result to semicomplete multipartite digraphs with 2-kings replaced by 4-kings is not true. In this paper, we (1) characterize all semicomplete digraphs D which are contained in a semicomplete multipartite digraph whose 4-kings are exactly the vertices of D. While it is trivial that K 4 ( Q ) ⊆ K 4 ( T ) , Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249–258] showed that K 3 ( Q ) ⊆ K 3 ( T ) and K 2 ( Q ) = K 2 ( T ) . Tan [On the kings and kings-of-kings in semicomplete multipartite digraphs, Discrete Math. 290 (2005) 249–258] also provided an example to show that K 3 ( Q ) need not be the same as K 3 ( T ) in general and posed the problem: characterize all those semicomplete multipartite digraphs T such that K 3 ( Q ) = K 3 ( T ) . In the course of proving our result (1), we (2) show that K 3 ( Q ) = K 3 ( T ) for all semicomplete multipartite digraphs T with no transmitters such that Q is a semicomplete digraph.