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Further Results on Existentially Closed Graphs Arising from Block Designs

  • Xiao-Nan LuEmail author
Original Paper
  • 7 Downloads

Abstract

A graph is n-existentially closed (n-e.c.) if for any disjoint subsets A, B of vertices with \(\left| {A \cup B} \right| =n\), 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 for such graphs to be 2-e.c. are established.

Keywords

Existential closure Block intersection graph Pairwise balanced design Triple system Steiner quadruple system 

Mathematics Subject Classification

05B07 05B05 05C75 

Notes

Acknowledgements

The author is grateful to the anonymous referee for the valuable suggestions and comments.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Industrial Administration, Faculty of Science and Technology Tokyo University of ScienceNoda-shiJapan

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