### Abstract

For a Steiner triple system of order v to have a complete s-arc one must have s(s + 1)/2≥v with equality only if s = 1 or 2 mod 4. To partition a Steiner triplesystem of order s(s + 1) 2 into complete s-arcs, one must have s = 1 mod 4. In this paper wegive constructions of Steiner triple systems of order s(s + 1) 2 which can be partitioned into complete s-arcs for all s = 1 mod 4. For s = 1 or 5 mod 12, we construct cyclic Steiner triple systems having this property. For s = 9 mod 12 we use Kirkman triple systems of order s having one additional property to construct these Steiner triple systems. We further establish that Kirkman triple systems having this additional property exist at least for s = 9 mod 24 and s = 21 mod 120.

Original language | English (US) |
---|---|

Pages (from-to) | 149-160 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 89 |

Issue number | 2 |

DOIs | |

State | Published - May 15 1991 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*89*(2), 149-160. https://doi.org/10.1016/0012-365X(91)90363-7

**Partitioning Steiner triple systems into complete arcs.** / Colbourn, Charles; Phelps, K. T.; de Resmini, M. J.; Rosa, A.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 89, no. 2, pp. 149-160. https://doi.org/10.1016/0012-365X(91)90363-7

}

TY - JOUR

T1 - Partitioning Steiner triple systems into complete arcs

AU - Colbourn, Charles

AU - Phelps, K. T.

AU - de Resmini, M. J.

AU - Rosa, A.

PY - 1991/5/15

Y1 - 1991/5/15

N2 - For a Steiner triple system of order v to have a complete s-arc one must have s(s + 1)/2≥v with equality only if s = 1 or 2 mod 4. To partition a Steiner triplesystem of order s(s + 1) 2 into complete s-arcs, one must have s = 1 mod 4. In this paper wegive constructions of Steiner triple systems of order s(s + 1) 2 which can be partitioned into complete s-arcs for all s = 1 mod 4. For s = 1 or 5 mod 12, we construct cyclic Steiner triple systems having this property. For s = 9 mod 12 we use Kirkman triple systems of order s having one additional property to construct these Steiner triple systems. We further establish that Kirkman triple systems having this additional property exist at least for s = 9 mod 24 and s = 21 mod 120.

AB - For a Steiner triple system of order v to have a complete s-arc one must have s(s + 1)/2≥v with equality only if s = 1 or 2 mod 4. To partition a Steiner triplesystem of order s(s + 1) 2 into complete s-arcs, one must have s = 1 mod 4. In this paper wegive constructions of Steiner triple systems of order s(s + 1) 2 which can be partitioned into complete s-arcs for all s = 1 mod 4. For s = 1 or 5 mod 12, we construct cyclic Steiner triple systems having this property. For s = 9 mod 12 we use Kirkman triple systems of order s having one additional property to construct these Steiner triple systems. We further establish that Kirkman triple systems having this additional property exist at least for s = 9 mod 24 and s = 21 mod 120.

UR - http://www.scopus.com/inward/record.url?scp=0041008672&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041008672&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(91)90363-7

DO - 10.1016/0012-365X(91)90363-7

M3 - Article

AN - SCOPUS:0041008672

VL - 89

SP - 149

EP - 160

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2

ER -