# (F, 2)—Rotational Steiner Triple Systems

• Zhike Jiang
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 329)

## Abstract

A Steiner triple system of order v is called (f, k)-rotational if it admits an automorphism consisting of f fixed points and k cycles of length (vf)/k. In this paper we deal with the existence of such Steiner triple systems when k = 2, f > 1 and show that an (f, 2)-rotational Steiner triple system of order v with f > 1 exists if and only if (i) v ≡ 1,3 (mod 6), (ii) f = 1,3 (mod 6), (iii) if vf ≡ 2 (mod 4) then v ≥ 3f; if vf = 0 (mod 4) then v = f or v ≥ 3f — 2, with the exceptions of v = 3f — 2 when f ≡ 1 (mod 12), v = 21 when f = 3, v = 3f — 2 or v = 3f + 10 when f ≡ 15 (mod 24), v = 3f + 4 when f = 3 (mod 12), v = 25 when f = 7, v = 3f – 2 when f ≡ 7 (mod 12), v = 27 when f = 9, v = 3f – 2 when f ≡ 21 (mod 24), v = 3f + 10 when f ≡ 9 (mod 24) and v = 3f + 4 when f = 9 (mod 12).

## Keywords

Discrete Math Triple System Existence Condition Projection Property Steiner Triple System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
C. J. Cho, Rotational Steiner triple systems, Discrete Math. 42 (1982), 153–159.
2. [2]
C. J. Colbourn and A. Rosa, Triple Systems, Oxford University Press, to appear.Google Scholar
3. [3]
J. Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1971–72), 315–319.
4. [4]
J. Doyen and R. M. Wilson, Embeddings of Steiner triple systems, Discrete Math. 5 (1973), 229–239.
5. [5]
R. B. Gardner, Steiner triple systems with near-rotational automorphisms, J. Combin. Theory, series A 61 (1992), 322–327.
6. [6]
A. Hartman and D. G. Hoffman, Steiner triple systems with an involution, Europ. J. Combin. 8 (1987), 371–378.
7. [7]
E. S. O’Keefe, Verification of a conjecture of Th. Skolem, Math Scand. 9 (1961), 80–82.
8. [8]
R. Peltesohn, Eine Losung der beiden Heffterschen Differenzenprobleme, Compositio Math. 6 (1939), 251–257.
9. [9]
K. T. Phelps and A. Rosa, Steiner triple systems with rotational automorphisms, Discrete Math. 33 (1981), 57–66.
10. [10]
A. Rosa, Poznámka o cyklických Steinerových systémoch trojic, Math.- Fyz. Čas 16 (1966), 285–290.
11. [11]
A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972), 61–71.
12. [12]
Th. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957), 57–68.
13. [13]
Th. Skolem, Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958), 273–280.
14. [14]
L. Teirlinck, The existence of reverse Steiner triple systems, Discrete Math. 6 (1973), 299–300.

## Authors and Affiliations

• Zhike Jiang
• 1
1. 1.Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada