Further Results on Existentially Closed Graphs Arising from Block Designs

TL;DR

This paper studies the properties of block intersection graphs derived from block designs, especially focusing on conditions under which these graphs are 2-existentially closed, with specific results for triple systems and Steiner quadruple systems.

Contribution

It provides new sufficient conditions for block intersection graphs of certain designs to be 2-e.c., advancing understanding of their structural properties.

Findings

Characterized when block intersection graphs of $\lambda$-fold triple systems are 1- or 2-e.c.

Established necessary and sufficient conditions for Steiner quadruple systems to have 2-e.c. block intersection graphs.

Proposed a sufficient condition for block intersection graphs of pairwise balanced designs to be 2-e.c.

Abstract

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ 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 $\cal B$, its block intersection graph is the graph whose vertex set is $\cal B$ 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 for such graphs to be $2$-e.c. are established.