### Abstract

A spanning set in a Steiner triple system is a set of elements for which each element not in the spanning set appears in at least one triple with a pair of elements from the spanning set. A scattering set is a set of elements that is independent, and for which each element not in the scattering set is in at most one triple with a pair of elements from the scattering set. For each v ≡ 1, 3 (mod 6), we exhibit a Steiner triple system with a spanning set of minimum cardinality, and a Steiner triple system with a scattering set of maximum cardinality. In the process, we establish the existence of Steiner triple systems with complete arcs of the minimum possible cardinality.

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

Pages (from-to) | 46-59 |

Number of pages | 14 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 57 |

Issue number | 1 |

DOIs | |

State | Published - 1991 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*57*(1), 46-59. https://doi.org/10.1016/0097-3165(91)90004-Z

**Spanning sets and scattering sets in Steiner triple systems.** / Colbourn, Charles; Dinitz, Jeffrey H.; Stinson, Douglas R.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 57, no. 1, pp. 46-59. https://doi.org/10.1016/0097-3165(91)90004-Z

}

TY - JOUR

T1 - Spanning sets and scattering sets in Steiner triple systems

AU - Colbourn, Charles

AU - Dinitz, Jeffrey H.

AU - Stinson, Douglas R.

PY - 1991

Y1 - 1991

N2 - A spanning set in a Steiner triple system is a set of elements for which each element not in the spanning set appears in at least one triple with a pair of elements from the spanning set. A scattering set is a set of elements that is independent, and for which each element not in the scattering set is in at most one triple with a pair of elements from the scattering set. For each v ≡ 1, 3 (mod 6), we exhibit a Steiner triple system with a spanning set of minimum cardinality, and a Steiner triple system with a scattering set of maximum cardinality. In the process, we establish the existence of Steiner triple systems with complete arcs of the minimum possible cardinality.

AB - A spanning set in a Steiner triple system is a set of elements for which each element not in the spanning set appears in at least one triple with a pair of elements from the spanning set. A scattering set is a set of elements that is independent, and for which each element not in the scattering set is in at most one triple with a pair of elements from the scattering set. For each v ≡ 1, 3 (mod 6), we exhibit a Steiner triple system with a spanning set of minimum cardinality, and a Steiner triple system with a scattering set of maximum cardinality. In the process, we establish the existence of Steiner triple systems with complete arcs of the minimum possible cardinality.

UR - http://www.scopus.com/inward/record.url?scp=0013543351&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013543351&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(91)90004-Z

DO - 10.1016/0097-3165(91)90004-Z

M3 - Article

AN - SCOPUS:0013543351

VL - 57

SP - 46

EP - 59

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -