On an adjacency property of almost all tournaments

Let n be a positive integer. A tournament is called n-existentially closed (or n-e.c.) if for every subset S of n vertices and for every subset T of S, there is a vertex x / S which is directed toward every vertex in T and directed away from every vertex in S\T . We prove that there is a 2-e.c. tournament with k vertices if and only if k 7 and k = 8, and give explicit examples for all such orders k. We also give a replication operation which preserves the 2-e.c. property.

• Type: Journal
• Year: 2006
• Where: DM
• Authors: Anthony Bonato, Kathie Cameron
• Keywords: DM 2006, Positive Integer, Replication Operation, Tournament