Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

A class of Steiner triple systems of order $ 21$ and associated Kirkman systems


Authors: Rudolf A. Mathon, Kevin T. Phelps and Alexander Rosa
Journal: Math. Comp. 37 (1981), 209-222
MSC: Primary 05B07; Secondary 51E10
Addendum: Math. Comp. 64 (1995), 1355-1356.
MathSciNet review: 616374
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Abstract: We examine a class of Steiner triple systems or order 21 with an automorphism consisting of three disjoint cycles of length 7. We exhibit explicitly all members of this class: they number 95 including the 7 cyclic systems. We then examine resolvability of the obtained systems; only 6 of the 95 are resolvable yielding a total of 30 nonisomorphic Kirkman triple systems of order 21. We also list several invariants of the systems and investigate their further properties.


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  • [1] W. Ahrens, Mathematische Unterhaltungen und Spiele, Bd. II, 2 Aufl., Teubner, Leipzig, 1918.
  • [2] László Babai, Almost all Steiner triple systems are asymmetric, Ann. Discrete Math. 7 (1980), 37–39. Topics on Steiner systems. MR 584402
  • [3] W. W. Rouse Ball, Mathematical Recreations and Essays, The Macmillan Company, New York, 1947. Revised by H. S. M. Coxeter. MR 0019629
  • [4] S. Bays, "Recherche des systèmes cycliques de triples de Steiner différents pour N premier (ou puissance de nombre premier)," J. Math. Pures Appl. (9), v. 2, 1923, pp. 73-98.
  • [5] A. Bray, "Twenty-one school-girl puzzle," Knowledge, v. 3, 1883, p. 268.
  • [6] Marlene J. Colbourn, An analysis technique for Steiner triple systems, Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), Congress. Numer., XXIII–XXIV, Utilitas Math., Winnipeg, Man., 1979, pp. 289–303. MR 561056
  • [7] Marlene J. Colbourn and Rudolf A. Mathon, On cyclic Steiner 2-designs, Ann. Discrete Math. 7 (1980), 215–253. Topics on Steiner systems. MR 584415
  • [8] F. N. Cole, Kirkman parades, Bull. Amer. Math. Soc. 28 (1922), no. 9, 435–437. MR 1560613, 10.1090/S0002-9904-1922-03599-9
  • [9] R. H. F. Denniston, "On the number of non-isomorphic reverse Steiner triple systems of order 19," Ann. Discrete Math., v. 7, 1980, pp. 255-264.
  • [10] Jean Doyen and Alexander Rosa, An extended bibliography and survey of Steiner systems, Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1977) Congress. Numer., XX, Utilitas Math., Winnipeg, Man., 1978, pp. 297–361. MR 535016
  • [11] H. E. Dudeney, "On Kirkman's schoolgirl problem," Educational Times Reprints (2), v. 15, 1909, pp. 17-19.
  • [12] Marshall Hall Jr., Combinatorial theory, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0224481
  • [13] R. Mathon and S. A. Vanstone, On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays, Discrete Math. 30 (1980), no. 2, 157–172. MR 566432, 10.1016/0012-365X(80)90117-X
  • [14] P. Mulder, Kirkman-systemen, Academisch Proefschrift ter verkrijging van den graad van doctor in de Wis- en Natuurkunde aan de Rijksuniversiteit te Groningen, Leiden, 1917.
  • [15] L. P. Petrenjuk & A. J. Petrenjuk, "Postroenie nekotoryh klassov kubičeskih grafov i neizomorfnost' Kirkmanovyh sistem troek," Kombin. Anal., v. 4, 1976, pp. 73-77.
  • [16] Kevin T. Phelps and Alexander Rosa, Steiner triple systems with rotational automorphisms, Discrete Math. 33 (1981), no. 1, 57–66. MR 597228, 10.1016/0012-365X(81)90258-2
  • [17] D. K. Ray-Chaudhuri and Richard M. Wilson, Solution of Kirkman’s schoolgirl problem, Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1971, pp. 187–203. MR 0314644
  • [18] A. Rosa, "Ispol'zovanie grafov dl'a rešenia zadači Kirkmana," Mat.-Fyz. Časopis, v. 13, 1963, pp. 105-113.
  • [19] Alexander Rosa, Generalized Howell designs, Second International Conference on Combinatorial Mathematics (New York, 1978), Ann. New York Acad. Sci., vol. 319, New York Acad. Sci., New York, 1979, pp. 484–489. MR 556058
  • [20] Alexander Rosa, Room squares generalized, Ann. Discrete Math. 8 (1980), 43–57. Combinatorics 79 (Proc. Colloq., Univ. Montréal, Montreal, Que., 1979), Part I. MR 597154
  • [21] Richard M. Wilson, Nonisomorphic Steiner triple systems, Math. Z. 135 (1973/74), 303–313. MR 0340046

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1981-0616374-9
Keywords: Steiner triple system, Kirkman triple system, isomorphism, resolvability
Article copyright: © Copyright 1981 American Mathematical Society