The King's Weigh House Lectures to Business Men.

On the 3-kings and 4-kings in multipartite tournaments

Koh and Tan gave a sufficient condition for a 3-partite tournament to have at least one 3-king in [K.M. Koh, B.P. Tan, Kings in multipartite tournaments, Discrete Math. 147 (1995) 171–183, Theorem 2]. In Theorem 1 of this paper, we extend this result to n-partite tournaments, where n 3. In [K.M. Koh, B.P. Tan, Number of 4-kings in bipartite tournaments with no 3-kings, Discrete Math. 154 (1996)...

Lectures

The Complexity of Kings

A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of t...

The Problem of the Kings

On a 2m 2n chessboard, the maximum number of nonattacking kings that can be placed is mn, since each 22 cell can have at most one king. Let f m (n) denote the number of ways that mn nonattacking kings can be placed on a 2m 2n chessboard. The purpose of this paper is to prove the following result. such that f m (n) = (c m n + d m)(m + 1) n + O(n m) (n ! 1): For every such placement of kings, the...

On the Domination of Kings

In this paper, we consider counting the number of ways to place kings on an k × n chessboard, such that every square is dominated by a king. Let f(k, n) be the number of dominating configurations. We consider the asymptotic behavior of the function f(k, n).