Kings in multipartite tournaments

Let T be an n-partite tournament and let k(r)(T) denote the number of r-kings of T. Gutin (1986) and Petrovic and Thomassen (1991) proved independently that if T contains at most one transmitter, then k(4)(T) greater than or equal to 1, and found infinitely many bipartite tournaments T with at most one transmitter such that k(3)(T) = 0. In this paper, we (i) obtain some sufficient conditions for T to have k(3)(T) greater than or equal to 1, (ii) show that if T contains no transmitter, then k(4)(T) greater than or equal to 4 when n = 2, and k(4)(T) greater than or equal to 3 when n greater than or equal to 3, and (iii) characterize all T with no transmitter such that the equalities in (ii) hold.