### Abstract

A (K_{4} - e)-design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v / 3, and that when equality is met, such a minimum embedding of an STS(v) exists, except when v = 15. Equality only holds when v ≡ 15, 27 (mod 30). One natural question is: What is the smallest order w such that some STS(v) can be embedded into a (K_{4} - e)-design of order v + w? We solve the problem for 7 of the 10 congruence classes modulo 30.

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

Pages (from-to) | 400-411 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 309 |

Issue number | 2 |

DOIs | |

State | Published - Jan 28 2009 |

### Fingerprint

### Keywords

- Embedding

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

_{4}- e)-designs II.

*Discrete Mathematics*,

*309*(2), 400-411. https://doi.org/10.1016/j.disc.2007.12.026

**Minimum embedding of Steiner triple systems into (K _{4} - e)-designs II.** / Ling, Alan C H; Colbourn, Charles; Quattrocchi, Gaetano.

Research output: Contribution to journal › Article

_{4}- e)-designs II',

*Discrete Mathematics*, vol. 309, no. 2, pp. 400-411. https://doi.org/10.1016/j.disc.2007.12.026

_{4}- e)-designs II. Discrete Mathematics. 2009 Jan 28;309(2):400-411. https://doi.org/10.1016/j.disc.2007.12.026

}

TY - JOUR

T1 - Minimum embedding of Steiner triple systems into (K4 - e)-designs II

AU - Ling, Alan C H

AU - Colbourn, Charles

AU - Quattrocchi, Gaetano

PY - 2009/1/28

Y1 - 2009/1/28

N2 - A (K4 - e)-design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v / 3, and that when equality is met, such a minimum embedding of an STS(v) exists, except when v = 15. Equality only holds when v ≡ 15, 27 (mod 30). One natural question is: What is the smallest order w such that some STS(v) can be embedded into a (K4 - e)-design of order v + w? We solve the problem for 7 of the 10 congruence classes modulo 30.

AB - A (K4 - e)-design of order v + w embeds a given Steiner triple system if there is a subset of v points on which the graphs of the design induce the blocks of the original Steiner triple system. It has been established that w ≥ v / 3, and that when equality is met, such a minimum embedding of an STS(v) exists, except when v = 15. Equality only holds when v ≡ 15, 27 (mod 30). One natural question is: What is the smallest order w such that some STS(v) can be embedded into a (K4 - e)-design of order v + w? We solve the problem for 7 of the 10 congruence classes modulo 30.

KW - Embedding

UR - http://www.scopus.com/inward/record.url?scp=56649109589&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56649109589&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2007.12.026

DO - 10.1016/j.disc.2007.12.026

M3 - Article

AN - SCOPUS:56649109589

VL - 309

SP - 400

EP - 411

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2

ER -