#### DMCA

## Independent sets in Steiner triple systems

Venue: | Ars Comb |

Citations: | 2 - 1 self |

### Citations

127 | Triple Systems
- Colbourn, Rosa
- 1999
(Show Context)
Citation Context ... implementation of Stinson’s hill-climbing algorithm [8], it is not too difficult to construct for each k ∈ {8, 9, 10} an STS(21) having an independent set of maximum cardinality k (Colbourn and Rosa =-=[1]-=-, Section 17.2). The only remaining possibility is that there might exist an STS(21) whose largest independent set has fewer than eight points. However, once we have obtained the formula for I8(v) we ... |

29 |
Hill-Climbing Algorithm for the Construction of Combinatorial Designs
- Stinson
(Show Context)
Citation Context ..., 3 ≤ k ≤ 8. Sauer and Schönheim [7] show that an STS(21) cannot have an independent set of cardinality greater than ten. Moreover, with a suitable implementation of Stinson’s hill-climbing algorithm =-=[8]-=-, it is not too difficult to construct for each k ∈ {8, 9, 10} an STS(21) having an independent set of maximum cardinality k (Colbourn and Rosa [1], Section 17.2). The only remaining possibility is th... |

22 |
Mendelsohn: A small basis for fourline configurations in Steiner triple systems
- Grannell, Griggs, et al.
- 1995
(Show Context)
Citation Context ...Indeed, the first is just the formula for the number of blocks in an STS(v): a0 = 1 6 v(v − 1), a1 = nv 72 (v − 7), a2 = nv 8 . The next nine equalities are taken from Grannell, Griggs and Mendelsohn =-=[4]-=-: b2 = nv 48 (v − 7)(v − 9), b3 = nv 48 (v − 5), b4 = nv 8 (v − 7), b5 = nv 6 , c10 = nv 8 (v − 8) + 3p, c11 = nv 4 (v − 7), c12 = nv 4 (v − 9) + 12p, c14 = nv 4 − 6p, c15 = nv 6 . 3sThe formulae for ... |

12 |
Counting frequencies of configurations in Steiner triple systems, Ars Combinatoria 46
- Horak, Phillips, et al.
- 1997
(Show Context)
Citation Context ...ine of them: C16, D1, E1, E2, E3, F1, F2, F3, G1, (1) and they are all variable. The main reason for our interest in these configurations is a theorem established by Horák, Phillips, Wallis and Yucas =-=[6]-=-: Any constant n-block configuration, together with all m-block configurations for m ≤ n having all points of degree at least two form a generating set for the n-line configurations. This theorem guar... |

10 |
Maximal subsets of a given set having no triple in common with a Steiner triple system on the set
- Sauer, Schönheim
- 1969
(Show Context)
Citation Context ...or Ik(S) in terms of the numbers of occurrences in S of certain configurations. This is stated as Theorem 1. From this formula we obtain explicit expressions for Ik(v), 3 ≤ k ≤ 8. Sauer and Schönheim =-=[7]-=- show that an STS(21) cannot have an independent set of cardinality greater than ten. Moreover, with a suitable implementation of Stinson’s hill-climbing algorithm [8], it is not too difficult to cons... |

8 |
On the chromatic numbers of Steiner triple systems
- Haddad
- 1999
(Show Context)
Citation Context ... 2 then allows us to solve the problem of determining those values of χ for which there exists an STS(21) with chromatic number χ. It is well known that STS(21)s with chromatic number 3 exist. Haddad =-=[5]-=- constructs an STS(21) with chromatic number 4, and Forbes, Grannell and Griggs [3] prove that every STS(21) is 5-colourable. Using Theorem 2 it is relatively straightforward to improve this last resu... |

7 |
Five-line configurations in Steiner triple systems
- Danziger, Mendelsohn, et al.
- 1996
(Show Context)
Citation Context ...1 configurations of at most eight points, and for convenience we list them in Table 1. For brevity, set brackets and commas have been omitted. The numbering assigned to the configurations is standard =-=[1, 2]-=-. Of particular relevance are configurations in the table that have no points of degree 1. There are precisely nine of them: C16, D1, E1, E2, E3, F1, F2, F3, G1, (1) and they are all variable. The mai... |

6 | On colourings of Steiner triple systems
- Forbes, Grannell, et al.
(Show Context)
Citation Context ...here exists an STS(21) with chromatic number χ. It is well known that STS(21)s with chromatic number 3 exist. Haddad [5] constructs an STS(21) with chromatic number 4, and Forbes, Grannell and Griggs =-=[3]-=- prove that every STS(21) is 5-colourable. Using Theorem 2 it is relatively straightforward to improve this last result and thereby completely determine the spectrum of chromatic numbers for STS(21)s.... |