### 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) |
---|---|

Pages (from-to) | 827-834 |

Number of pages | 8 |

Journal | Discrete Mathematics |

Volume | 311 |

Issue number | 10-11 |

DOIs | |

State | Published - Jun 6 2011 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

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

**Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19.** / Colbourn, Charles; Kaski, Petteri; Stergrd, Patric R J; Pike, David A.; Pottonen, Olli.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 311, no. 10-11, pp. 827-834. https://doi.org/10.1016/j.disc.2011.02.005

}

TY - JOUR

T1 - Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19

AU - Colbourn, Charles

AU - Kaski, Petteri

AU - Stergrd, Patric R J

AU - Pike, David A.

AU - Pottonen, Olli

PY - 2011/6/6

Y1 - 2011/6/6

N2 - A Hanani triple system of order 6n+1, HATS(6n+1), is a decomposition of the complete graph K6n+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 K6n-F into 3n-1 sets of 2n disjoint triangles; here F is a one-factor of K6n. 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.

AB - A Hanani triple system of order 6n+1, HATS(6n+1), is a decomposition of the complete graph K6n+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 K6n-F into 3n-1 sets of 2n disjoint triangles; here F is a one-factor of K6n. 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.

KW - Hanani triple system

KW - Nearly Kirkman triple system

KW - Resolvable design

KW - Steiner triple system

UR - http://www.scopus.com/inward/record.url?scp=79951943756&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951943756&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2011.02.005

DO - 10.1016/j.disc.2011.02.005

M3 - Article

AN - SCOPUS:79951943756

VL - 311

SP - 827

EP - 834

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 10-11

ER -