Further Results on Existentially Closed Graphs Arising from Block Designs

Xiao-Nan Lu

Published 2018-10-17Version 1

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint vertex $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block set $\mathcal{B}$, its block intersection graph is the graph whose vertex set is $\mathcal{B}$ 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 $\lambda$-fold triple systems with $\lambda \ge 2$ 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 when such graphs are $2$-e.c. are established.