Number of 4-kings in bipartite tournaments with no 3-kings (Q1918562)

If \(T\) is an oriented graph, let \(k_j(T)\) denote the number of nodes \(u\) of \(T\) such that the distance from \(u\) to any other node of \(T\) is at most \(j\). The authors show that if \(T\) is a bipartite tournament such that \(T\) has no nodes of in-degree zero and \(k_3(T)= 0\), then \(k_4(T)\geq 8\); and they characterize the tournaments for which equality holds.