Abstract
We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs.
Highlights
In the paper we investigate the structure which yield complete graphs contained in a Steiner triple system
In what follows, speaking about a graph G contained in a configuration M we always assume that G is freely contained in M i.e. it is not merely a subgraph of the collinearity graph of M and distinct edges of
We give only a technique to construct a binomial configuration which contains at least the given number of Kn-subgraphs: Theorem 3.12, and we prove that for each admissible integer m there does exist a binomial configuration which freely contains m Kn-subgraphs
Summary
In the paper we investigate the structure which (may) yield complete graphs contained in a (partial) Steiner triple system (in short: in a PSTS). In our case configurations in question are complete graphs and so-called binomial configurations. G lie on distinct lines (so called sides) of M, and sides do not intersect outside G Note that following this terminology a complete quadrangle on a projective plane P is not a K4-graph contained in P. -configuration M may contain follows: this size is n − 1 In this case we say that Kn is a maximal (complete) subgraph of M. Binomial configurations with the maximal number of Kn-subgraphs are exactly the generalized Desargues configurations. Some remarks are made which show that, surprisingly, binomial configurations with ‘many’ maximal complete subgraphs (the maximal admissible number, the maximal number reduced by 2, and the maximal number reduced by 3) are the configurations of some well known classes
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