### Abstract

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (v 2) - (u 2) - (w 2) a? 0 (mod3), and v a?Y w + u + max u,w . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v - u - w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well-known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem

Original language | English |
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Pages (from-to) | 14 - 24 |

Number of pages | 11 |

Journal | Journal of Combinatorial Designs |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 |

Externally published | Yes |

## Cite this

Bryant, D., & Horsley, D. (2006). Steiner triple systems with two disjoint subsystems.

*Journal of Combinatorial Designs*,*14*(1), 14 - 24. https://doi.org/10.1002/jcd.20071