Embed Size (px)
Citation preview
Characterization of affine Steiner triple systemsand Hall triple systems
Daniel Kral’ 1,4
Institute for Theoretical Computer Science (ITI)Faculty of Mathematics and Physics, Charles UniversityMalostranske namestı 25, 118 00 Prague, Czech Republic
Edita Macajova 2,5
Department of Computer ScienceFaculty of Mathematics, Physics and Informatics, Comenius University
Mlynska dolina, 842 48 Bratislava, Slovakia
Attila Por 6
Department of Applied MathematicsFaculty of Mathematics and Physics, Charles UniversityMalostranske namestı 25, 118 00 Prague, Czech Republic
Jean-Sebastien Sereni 3,7
Department of Applied MathematicsFaculty of Mathematics and Physics, Charles UniversityMalostranske namestı 25, 118 00 Prague, Czech Republic
Abstract
It is known that a Steiner triple system is projective if and only if it does not containthe four-triple configuration C14. We find three configurations such that a Steiner
Electronic Notes in Discrete Mathematics 29 (2007) 17–21
1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
www.elsevier.com/locate/endm
doi:10.1016/j.endm.2007.07.004
triple system is affine if and only if it does not contain any of these configurations.Similarly, we characterize Hall triple systems, a superclass of affine Steiner triplesystems, using two forbidden configurations.
Keywords: Steiner triple systems, Hall triple systems.
1 Introduction
Steiner triple systems form a classical notion in combinatorial design theory.A Steiner triple system S is formed by n points and several triples such thatevery two distinct points are contained in exactly one common triple. Inparticular, S has
(n
2
)/3 triples. One of the most classical results asserts the
existence of a Steiner triple system with n points whenever n = 1, 3 (mod 6),n ≥ 3. The amount of results on Steiner triple systems is enormous and aseparate monograph [1] on the topic has recently appeared.
There are several prominent classes of Steiner triple systems, e.g., pro-jective and affine Steiner triple systems. A projective Steiner triple system
PG(d, 2) is the Steiner triple system with 2d+1 − 1 points corresponding tonon-zero (d + 1)-dimensional vectors over Z2 for d ≥ 1. Three such vectorsform a triple if they sum to the zero vector. The smallest Steiner triple sys-tem is the projective system PG(1, 2), the system comprised of three pointsforming a single triple, also referred to as the trivial Steiner triple system.The smallest non-trivial projective Steiner triple system S7 is PG(2, 2), theFano plane—the unique Steiner triple system with seven points. An affine
Steiner triple system AG(d, 3) is the system with 3d points corresponding tod-dimensional vectors over Z3 for d ≥ 1. Three such vectors form a triple ofAG(d, 3) if they sum to the zero vector. The smallest non-trivial affine Steinertriple system S9 is AG(2, 3), the unique Steiner triple system with nine points.
It is natural to ask whether these two classes of Steiner triple systemscan be characterized in terms of well-described forbidden substructures (asfor instance, it is known that a graph is planar if and only if it does not
1 Institute for Theoretical computer science is supported as project 1M0545 by CzechMinistry of Education.2 This research was supported by APVT project no. 51-027604.3 This author has been supported by EU project Aeolus.4 Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected]
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 17–2118
Fig. 1. The configuration C14 and the Pasch configuration C16.
contain a subdivision of one of the graphs K3,3 or K5). To be more precise, aconfiguration C is formed by points and triples such that each pair of pointsis in at most one of the triples, and a Steiner triple system S contains C ifthere is an injective mapping of the points of C to the points of S such thattriples of points of C are mapped to triples of points of S.
The solution to the just stated problem is easy to find for the class ofprojective Steiner triple systems. Two four-triple configurations play impor-tant roles in this characterization: the first one is the configuration C14 andthe second one is the configuration C16, known as the Pasch configuration
(see Figure 1). The configuration C14 cannot be contained in any projectiveSteiner triple system, and the converse is also true: Stinson and Wei [6] es-tablished that a Steiner triple system S with n points is projective if and onlyif it contains 1
24n(n − 1)(n − 3) distinct copies of the Pasch configuration.
By a counting argument given by Grannel et al. [2], if S contains less than1
24n(n − 1)(n − 3) copies of the Pasch configuration, then it must contain a
configuration isomorphic to C14. We state this observation as a theorem.
Theorem 1.1 (Grannel et al. [2] and Stinson et al. [6]) A Steiner triple
system S is projective if and only if it contains no configuration C14.
However, we were not able to find such a simple argument characterizingaffine Steiner triple systems in the literature.
A finer distinction between affine and non-affine Steiner triple systems canbe achieved. A Steiner triple system S is a Hall triple system if for every pointx of S, there exists an automorphism of S that is an involution and its onlyfixed point is x. Hall [3] showed that a Steiner triple system is a Hall triplesystem if and only if every Steiner triple system induced by the points of twonon-disjoint triples of S is isomorphic to S9. Recall that the Steiner triplesystem induced by a set X of the points of S is the smallest Steiner triplesystem S ′ such that S ′ contains all the points of X and all triples of S ′ arealso triples of S. Hence, Hall triple systems look “locally” as affine Steinertriple systems and it can seem hard to distinguish these two classes in termsof forbidden substructures. Also note that if any two non-disjoint triples of
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 17–21 19
Fig. 2. The three forbidden configuration for affine Steiner triple systems: theconfigurations C16, C1
S and C2S (from left to right).
Fig. 3. The anti-mitre configuration CA.
S induce a Steiner triple system isomorphic to S7, then S must be projectiveunlike in the case of affine Steiner triple systems (there are examples of Halltriple systems with n points that are not affine for every n = 3d, d ≥ 4).
2 Our results
We show that a Steiner triple system is affine unless it contains one of thethree configurations depicted in Figure 2, the Pasch configuration C16, theconfiguration C1
S and the configuration C2S.
Theorem 2.1 A Steiner triple system S is an affine Steiner triple system if
and only if it contains none of the configurations C16, C1S and C2
S.
Observe now that no Hall triple system contains the configuration C16 orthe anti-mitre configuration CA (see Figure 3) as neither of them is containedin the Steiner triple system S9. The converse is also true.
Theorem 2.2 A Steiner triple system S is a Hall triple system if and only if
it does not contain the configuration C16 or CA.
Theorem 2.3 A Hall triple system is affine if and only if it contains none of
the configurations C1S and C2
S.
If we additionally assume that a Steiner triple system S is not projective,we can remove the configuration C16 from the list of forbidden configurationsboth for affine Steiner triple systems and Hall triple systems (Theorems 2.1
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 17–2120
and 2.2 are implied by these stronger results). Let us point out that everySteiner triple system containing the configuration CA also contains one of theconfigurations C1
S and C2S (this seems to be a non-trivial fact).
Our characterization results have several interesting corollaries in the areaof edge-colorings of graphs [5]. A cubic graph G is S-edge-colorable for aSteiner triple system S if its edges can be colored with points of S in such away that the points assigned to three edges sharing a vertex form a triple inS. Among others, we can establish using these results the following conjectureof Holroyd and Skoviera [4] for point-transitive Steiner triple systems:
Conjecture 2.4 (Holroyd and Skoviera [4], Conjecture 1.4) Let S be a
non-projective Steiner triple system. A cubic graph G is S-edge-colorable un-
less G has a bipartite end and S is affine.
Acknowledgement
The second author would like to thank Department of Applied Mathematics ofCharles University for hospitality during her stay in Prague when this researchwas conducted. All the authors are indebted to Alexander Rosa for his insightson Steiner triple systems, in particular, for pointing out the counting argumentthat C14 is a forbidden configuration for projective Steiner triple systems.
References
[1] C. J. Colbourn, and A. Rosa, “Triple systems”, Oxford Univ. Press, Oxford,1999.
[2] M. J. Grannel, T. S. Griggs, and E. Mendelsohn, A small basis for four-lineconfigurations in Steiner triple systems, J. Combin. Des. 3 (1995), 51–59.
[3] M. Hall, Jr., Automorphisms of Steiner triple systems, IBM J. Res. Develop.4(5) (1960), 460–472.
[4] F. C. Holroyd, and M. Skoviera, Colouring of cubic graphs by Steiner triplesystems, J. Combin. Theory Ser. B. 91 (2004), 57–66.
[5] D. Kral’, E. Macajova, A. Por, and J.-S. Sereni, Characterization results forSteiner triple systems and their application to edge-colorings of cubic graphs, inpreparation.
[6] D. R. Stinson, and Y. J. Wei, Some results on quadrilaterals in Steiner triplesystems, Discrete Math. 105 (1992), 207–219.
D. Král’ et al. / Electronic Notes in Discrete Mathematics 29 (2007) 17–21 21
