Abstract :
Let T be an n-partite tournament and let kr(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 k4(T) ⩾ 1, and found infinitely many bipartite tournaments T with at most one transmitter such that k3 (T) = 0. In this paper, we (i) obtain some sufficient conditions for T to have k3 (T) ⩾ 1, (ii) show that if T contains no transmitter, then k4 (T) ⩾ 4 when n = 2, and k4 (T) ⩾ 3 when n ⩾ 3, and (iii) characterize all T with no transmitter such that the equalities in (ii) hold.
