The spectrum of 1-rotational Steiner triple systems over a dicyclic group

The spectrum of values v for which a 1-rotational Steiner triple system of order v exists over a dicyclic group is determined.     

The spectrum for rotational Steiner triple systems

We develop some recursive constructions for rotational Steiner triple systems with which the spectrum of a k-rotational Steiner triple system of order v is completely determined for each positive integer k. © 1996 John Wiley & Sons, Inc.     

Some progress on the existence of 1-rotational Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57–66, 1981), Buratti (J Combin Des 9:215–226, 2001) and Mishima (Discrete Math 308:2617–2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v =  (p ³−p)n +  1 ≡ 1 (mod 96) with p a prime, n \not o 0{n \not\equiv 0} (mod 4), and the odd part of (p ³ − p)n that is square-free and without prime factors congruent to 1 (mod 6).     

A Note on Partial Parallel Classes in Steiner Systems

A partial parallel class of blocks of a Steiner system S(t,k,v) is a collection of pairwise disjoint blocks. The purpose of this note is to show that any S(k,k+1,v) Steiner system, with v?k⁴+3k³+k²+1, has a partial parallel class containing at least (v?k+1)/(k+2) blocks.     

Concerning multiplier automorphisms of cyclic Steiner triple systems

A cyclic Steiner triple system, presented additively over Z _v as a set B of starter blocks, has a non-trivial multiplier automorphism λ ≠ 1 when λB is a set of starter blocks for the same Steiner triple system. When does a cyclic Steiner triple system of order v having a nontrivial multiplier automorphism exist? Constructions are developed for such systems; of most interest, a novel extension of Netto's classical construction for prime orders congruent to 1 (mod 6) to prime powers is proved. Nonexistence results are then established, particularly in the cases when v = (2^β + 1)^α, when v = 9p with p ≡ 5 (mod 6), and in certain cases when all prime divisors are congruent to 5 (mod 6). Finally, a complete solution is given for all v < 1000, in which the remaining cases are produced by simple computations.     

On steiner spaces

A Steiner space is a Steiner triple system that is not generated by a triangle. We give new constructions of Steiner spaces and solve the existence problem of Steiner spaces of order v for all but 4 values of v.     

The existence of resolvable Steiner quadruple systems

A Steiner quadruple system of order v is a set X of cardinality v, and a set Q, of 4-subsets of X, called blocks, with the property that every 3-subset of X is contained in a unique block. A Steiner quadruple system is resolvable if Q can be partitioned into parallel classes (partitions of X). A necessary condition for the existence of a resolvable Steiner quadruple system is that v≡4 or 8 (mod 12). In this paper we show that this condition is also sufficient for all values of v, with 24 possible exceptions.     

On large sets of disjoint steiner triple systems,VI

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that D(v) = v ? 2 holds for all v ≡ 1, 3 (mod 6) (v>7), except possibly v = 141, 283, 501, 789, 1501, 2365.     

Nonexistence of sparse triple systems over abelian groups and involutions

In 1973 Paul Erdős conjectured that there is an integer v ₀(r) such that, for every v>v ₀(r) and v≡1,3 (mod 6), there exists a Steiner triple system of order v, containing no i blocks on i+2 points for every 1<i≤r. Such an STS is said to be r-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every r≥13 there exists no point-transitive r-sparse STS over an abelian group. This bound and the classification of transitive groups give further nonexistence results on block-transitive, flag-transitive, 2-transitive, and 2-homogeneous STSs with high sparseness. We also give stronger bounds on the sparseness of STSs having some particular automorphisms with small groups. As a corollary of these results, it is shown that various well-known automorphisms, such as cyclic, 1-rotational over arbitrary groups, and involutions, prevent an STS from being high-sparse.      

Some packings with Steiner triple systems

For eleven values of v, the following problem is solved To partition, into v?2 Steiner triple system, the set of all the combinations of v things three at a time.     

On the maximum number of disjoint Steiner triple systems

Let D(v) be the maximum number of pairwise disjoint Steiner triple systems of order v. We prove that D(3v)≥2v+D(v) for every v ≡ 1 or 3 (mod 6), v≥3. As a corollary, we have D(3ⁿ)=3ⁿ-2 for every n≥1.     

A theorem on the maximum number of disjoint steiner triple systems

We prove that D(2v + 1) ? v + 1 + D(v) for v > 3 where D(v) denotes the maximum number of pairwise disjoint Steiner triple systems of order v. Since D(v) ? v ? 2 it follows that for v > 3, D(2v + 1) = 2v ?1 whenever D(v) = v ? 2.     

Super-simple Steiner pentagon systems

A Steiner pentagon system of order v(SPS(v)) is said to be super-simple if its underlying (v,5,2)-BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary condition for the existence of a super-simple SPS(v); namely, v?5 and v≡1 or is sufficient, except for v=5, 15 and possibly for v=25. In the process, we also improve an earlier result for the spectrum of super-simple (v,5,2)-BIBDs, removing all the possible exceptions. We also give some new examples of Steiner pentagon packing and covering designs (SPPDs and SPCDs).     

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ^?1-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ^?1-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2^?1-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2^?1-isomorphic and 3^?1-isomorphic pairs of STS(15)s.     

Super-Simple Twofold Steiner Pentagon Systems

A twofold pentagon system of order v is a decomposition of the complete undirected 2-multigraph 2K _v into pentagons. A twofold Steiner pentagon system of order v [TSPS(v)] is a twofold pentagon system such that every pair of distinct vertices is joined by a path of length two in exactly two pentagons of the system. A TSPS(v) is said to be super-simple if its underlying (v, 5, 4)-BIBD is super-simple; that is, if any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary conditions for the existence of a super-simple TSPS(v); namely, v ≥ 15 and v ≡ 0 or 1 (mod 5) are sufficient. For these specified orders, the main result of this paper also guarantees the existence of a very special and interesting class of twofold and fourfold Steiner pentagon systems of order v with the additional property that, for any two vertices, the two or four paths of length two joining them are distinct.     

Halving Steiner 2-designs

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into K_ks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k?5 or any Mersenne prime k, there is a constant number v₀ such that if v>v₀ and v satisfies the above necessary condition, then there exists a halvable S(2,k,v). We also show that a halvable S(2,2ⁿ,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.     

On making two Steiner triple systems disjoint

We prove that if (S₁, β₁) and (S₂, β₂) are two Steiner triple systems of order v and if S is a set of v points, then there exist two disjoint Steiner triple systems (S, β₁′) and (S, β₂′) with (S₁, β₁) ? (S, β₁′) and (S₂, β₂) ? (S, β₂′).     

On large sets of disjoint Steiner triple systems II

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that if n is an odd number, there exist 12 mutually orthogonal Latin squares of order n and D(1 + 2n) = 2n ? 1, then D(1 + 12n) = 12n ? 1.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

On large sets of disjoint steiner triple systems I

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, it is proved that if D(2 + n) = n, p is a prime number, p ≡ 7 (mod 8) or p? {5, 17, 19, 2}, and (p, n) ≠ (5, 1), then D(2 + pn) = pn.