On an adjacency property of almost all graphs

A graph is called n-existentially closed or n-e.c. if it satis/es the following adjacency property: for every n-element subset S of the vertices, and for every subset T of S, there is a vertex not in S which is joined to all of T and to none of S\T . The unique countable random graph is known to be n-e.c. for all n. Equivalently, for any /xed n, almost all /nite graphs are n-e.c. However, few examples of n-e.c. graphs are known other than large Paley graphs and examples of 2-e.c. graphs given in (Cacetta, et al., Ars Combin. 19 (1985) 287–294). An n-e.c. graph is critical if deleting any vertex leaves a graph which is not n-e.c. We classify the 1-e.c. critical graphs. We construct 2-e.c. critical graphs of each order ¿ 9, and describe a 2-e.c.-preserving operation: replication of an edge. We also examine which of the well-known binary operations on graphs preserve n-e.c. for n=1; 2; 3. c © 2001 Elsevier Science B.V. All rights reserved. MSC: 05C35; 05C80; 05C75