Toppling kings in multipartite tournaments by introducing new kings

Abstract

Let t = ( t 1 , t 2 , … , t n ) and c = ( c 1 , c 2 , … , c n ) be two n -tuples of nonnegative integers. An all-4-kings n -partite tournament T ( V 1 , V 2 , … V n ) is said to have a ( t , c ) - property if there exists an n -partite tournament T 1 ( W 1 , W 2 , … , W n ) such that for each i ∈ { 1 , … , n } : (1) V i ⊆ W i ; (2) exactly t i 4-kings of V i are not 4-kings in T 1 ; (3) exactly c i 4-kings of W i are not vertices of V i . We describe all pairs ( t , c ) such that there exists an n -partite tournament having ( t , c ) -property.