Further Results on Existentially Closed Graphs Arising from Block Designs.

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with A ∪ B = n , there is a vertex z ∉ A ∪ B adjacent to every vertex of A and no vertex of B. For a block design with block set B , its block intersection graph is the graph whose vertex set is B and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be 2-e.c. In particular, we study the λ -fold triple systems with λ ≥ 2 and determine for which parameters their block intersection graphs are 1- or 2-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called { 1 } -block intersection graphs are investigated, and the necessary and sufficient conditions for such graphs to be 2-e.c. are established. [ABSTRACT FROM AUTHOR]