Existentially closed BIBD block-intersection graphs
Summary: A graph \(G\) with vertex set \(V\) is said to be \(n\)-existentially closed if, for every \(S \subset V\) with \(|S|=n\) and every \(T \subseteq S\), there exists a vertex \(x \in V-S\) such that \(x\) is adjacent to each vertex of \(T\) but is adjacent to no vertex of \(S-T\). Given a combinatorial design \({\mathcal D}\) with block set \({\mathcal B}\), its block-intersection graph \(G_{\mathcal D}\) is the graph having vertex set \({\mathcal B}\) such that two vertices \(b_1\) and \(b_2\) are adjacent if and only if \(b_1\) and \(b_2\) have non-empty intersection. In this paper we study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are \(n\)-existentially closed. We characterise the BIBDs with block size \(k \geq 3\) and index \(\lambda=1\) that have 2-e.c.\ block-intersection graphs and establish bounds on the parameters of BIBDs with index \(\lambda=1\) that are \(n\)-e.c.\ where \(n \geq 3\). For \(\lambda \geq 2\) and \(n \geq 2\), we prove that only simple \(\lambda\)-fold designs can have \(n\)-e.c.\ block-intersection graphs. In the case of \(\lambda\)-fold triple systems we show that \(n \geq 3\) is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with \(k=3\) and \(\lambda=2\)) have 2-e.c.\ block-intersection graphs.
scientific article; zbMATH DE number 5540848