### Abstract

It has been shown that the number of occurrences of any ℓ-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full m-line configurations for 1 ≥ m ≥ ℓ full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in ν, the order of the Steiner triple system. Moreover, the counts of full configurations, together with ν, form a linear basis for all of the configuration counts when ℓ ≥7. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when ℓ = 6. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored.

Original language | English (US) |
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Pages (from-to) | 77-108 |

Number of pages | 32 |

Journal | Mathematica Slovaca |

Volume | 59 |

Issue number | 1 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### Keywords

- Configuration
- Polyhedral inequality
- R-sparsity
- Steiner triple system

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The configuration polytope of ℓ-line configurations in Steiner triple systems.** / Colbourn, Charles.

Research output: Contribution to journal › Article

*Mathematica Slovaca*, vol. 59, no. 1, pp. 77-108. https://doi.org/10.2478/s12175-008-0111-2

}

TY - JOUR

T1 - The configuration polytope of ℓ-line configurations in Steiner triple systems

AU - Colbourn, Charles

PY - 2009

Y1 - 2009

N2 - It has been shown that the number of occurrences of any ℓ-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full m-line configurations for 1 ≥ m ≥ ℓ full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in ν, the order of the Steiner triple system. Moreover, the counts of full configurations, together with ν, form a linear basis for all of the configuration counts when ℓ ≥7. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when ℓ = 6. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored.

AB - It has been shown that the number of occurrences of any ℓ-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full m-line configurations for 1 ≥ m ≥ ℓ full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in ν, the order of the Steiner triple system. Moreover, the counts of full configurations, together with ν, form a linear basis for all of the configuration counts when ℓ ≥7. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when ℓ = 6. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored.

KW - Configuration

KW - Polyhedral inequality

KW - R-sparsity

KW - Steiner triple system

UR - http://www.scopus.com/inward/record.url?scp=68149178829&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68149178829&partnerID=8YFLogxK

U2 - 10.2478/s12175-008-0111-2

DO - 10.2478/s12175-008-0111-2

M3 - Article

AN - SCOPUS:68149178829

VL - 59

SP - 77

EP - 108

JO - Mathematica Slovaca

JF - Mathematica Slovaca

SN - 0139-9918

IS - 1

ER -