### Abstract

A Hanani triple system of order 6n+1, HATS(6n+1), is a decomposition of the complete graph K_{6n+1} into 3n sets of 2n disjoint triangles and one set of n disjoint triangles. A nearly Kirkman triple system of order 6n, NKTS(6n), is a decomposition of K_{6n}-F into 3n-1 sets of 2n disjoint triangles; here F is a one-factor of K_{6n}. The Hanani triple systems of order 6n+1 and the nearly Kirkman triple systems of order 6n can be classified using the classification of the Steiner triple systems of order 6n+1. This is carried out here for n=3: There are 3787983639 isomorphism classes of HATS(19)s and 25328 isomorphism classes of NKTS(18)s. Several properties of the classified systems are tabulated. In particular, seven of the NKTS(18)s have orthogonal resolutions, and five of the HATS(19)s admit a pair of resolutions in which the almost parallel classes are orthogonal.

Original language | English (US) |
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Pages (from-to) | 827-834 |

Number of pages | 8 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 10-11 |

DOIs | |

State | Published - Jun 6 2011 |

### Keywords

- Hanani triple system
- Nearly Kirkman triple system
- Resolvable design
- Steiner triple system

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*311*(10-11), 827-834. https://doi.org/10.1016/j.disc.2011.02.005