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arXiv · 1810.07719

Further Results on Existentially Closed Graphs Arising from Block Designs

Xiao-Nan Lu
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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 $λ$-fold triple systems with $λ\ge 2$ 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.

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Xiao-Nan Lu. 2019-04-16. Further Results on Existentially Closed Graphs Arising from Block Designs. https://doi.org/10.1007/s00373-019-02036-z

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