Existence results for near resolvable designs

A near resolvable design, NRB(v, k), is a balanced incomplete block design whose block set can be partitioned into v classes such that each class contains every point of the design but one, and each point is missing from exactly one class. The necessary conditions for the existence of near resolvable designs are v ≡ 1 mod k and λ = k ? 1. These necessary conditions have been shown to be sufficient for k ? {2,3,4} and almost always sufficient for k ? {5,6}. We are able to show that there exists an integer n₀(k) so that NRB(v,k) exist for all v > n₀(k) and v ≡ 1 mod k. Using some new direct constructions we show that there are many k for which it is easy to compute an explicit bound on n₀(k). These direct constructions also allow us to build previously unknown NRB(v,5) and NRB(v,6). © 1995 John Wiley & Sons, Inc.     

On uniformly resolvable designs with block sizes 3 and 4

A Uniformly Resolvable Design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k -pc and for a given k the number of k -pcs is denoted r _k . In this paper we consider the case of block sizes 3 and 4. The cases r ₃ = 1 and r ₄ = 1 correspond to Resolvable Group Divisible Designs (RGDD). We prove that if a 4-RGDD of type h ^u exists then all admissible {3, 4}-URDs with 12hu points exist. In particular, this gives existence for URD with v ≡ 0 (mod 48) points. We also investigate the case of URDs with a fixed number of k -pc. In particular, we show that URDs with r ₃ = 4 exist, and that those with r ₃ = 7, 10 exist, with 11 and 12 possible exceptions respectively, this covers all cases with 1 < r ₃ ≤ 10. Furthermore, we prove that URDs with r ₄ = 7 exist and that those with r ₄ = 9 exist, except when v = 12, 24 and possibly when v = 276. In addition, we prove that there exist 4-RGDDs of types 2 ¹⁴², 2 ³⁴⁶ and 6 ⁵⁴. Finally, we provide four {3,5}-URDs with 105 points.     

Existence of resolvable H-designs with group sizes 2, 3, 4 and 6

In 1987, Hartman showed that the necessary condition v ≡ 4 or 8 (mod 12) for the existence of a resolvable SQS(v) is also sufficient for all values of v, with 23 possible exceptions. These last 23 undecided orders were removed by Ji and Zhu in 2005 by introducing the concept of resolvable H-designs. In this paper, we first develop a simple but powerful construction for resolvable H-designs, i.e., a construction of an RH(g ²ⁿ ) from an RH((2g) ⁿ ), which we call group halving construction. Based on this construction, we provide an alternative existence proof for resolvable SQS(v)s by investigating the existence problem of resolvable H-designs with group size 2. We show that the necessary conditions for the existence of an RH(2 ⁿ ), namely, n ≡ 2 or 4 (mod 6) and n ≥ 4 are also sufficient. Meanwhile, we provide an alternative existence proof for resolvable H-designs with group size 6. These results are obtained by first establishing an existence result for resolvable H-designs with group size 4, that is, the necessary conditions n ≡ 1 or 2 (mod 3) and n ≥ 4 for the existence of an RH(4 ⁿ ) are also sufficient for all values of n except possibly n ∈ {73, 149}. As a consequence, the general existence problem of an RH(g ⁿ ) is solved leaving mainly the case of g ≡ 0 (mod 12) open. Finally, we show that the necessary conditions for the existence of a resolvable G-design of type g ⁿ are also sufficient.     

On large sets of resolvable and almost resolvable oriented triple systems

An MTS(v) [or DTS(v)] is said to be resolvable, denoted by RMTS(v) [or RDTS(v)], if its block set can be partitioned into parallel classes. An MTS(v) [or DTS(v)] is said to be almost resolvable, denoted by ARMTS(v) [or ARDTS(v)], if its block set can be partitioned into almost parallel classes. The large set of RMTS(v) [or RDTS(v) or ARMTS(v) or ARDTS(v)] is denoted by LRMTS(v) [or LRDTS(v) or LARMTS(v) or LARDTS(v)]. In this article we do some preliminary study for their existence, and give several recursive theorems using other combinatorial structures. © 1996 John Wiley & Sons, Inc.     

The existence of doubly near resolvable (v,3,2)-BIBDs

The existence of doubly near resolvable (v,2,1)-BIBDs was established by Mullin and Wallis in 1975. In this article, we determine the spectrum of a second class of doubly near resolvable balanced incomplete block designs. We prove the existence of DNR(v,3,2)-BIBDs for v ≡ 1 (mod 3), v ≥ 10 and v ? {34,70,85,88,115,124,133,142}. The main construction is a frame construction, and similar constructions can be used to prove the existence of doubly resolvable (v,3,2)-BIBDs and a class of Kirkman squares with block size 3, KS₃(v,2,4). © 1994 John Wiley & Sons, Inc.     

On vector space partitions and uniformly resolvable designs

Let V _n (q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V _n (q) is a partition of V _n (q) if every nonzero vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

Octahedral designs

An oriented octahedral design of order v, or OCT(v), is a decomposition of all oriented triples on v points into oriented octahedra. Hanani [H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, 319, New York Academy of Science, New York, 1979, pp. 260–264.] settled the existence of these designs in the unoriented case. We show that an OCT(v) exists if and only if v≡1, 2, 6 (mod 8) (the admissible numbers), and moreover the constructed OCT(v) are unsplit, i.e. their octahedra cannot be paired into mirror images. We show that an OCT(v) with a subdesign OCT(U) exists if and only if v and u are admissible and v≥u+4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:319–327, 2010     

Class-uniformly resolvable designs with block sizes 2 and 3

In this article we consider restricted resolvable designs (RRP) with block sizes 2 and 3 that are class uniform. A characterization scheme is developed, based on the ratio a:b of pairs to triples, and necessary conditions are provided for the existence of these designs based on this characterization. We show asymptotic existence results when (a,b) = (1, 2n), n ≥ 1 and when (a,b) = (9, 2). We also study the specific cases when (a,b) = (1, 2n), 1 ≤ n ≤ 5, (a,b) = (3, 6u−2), u ≥ 1 and when (a,b) E {(1, 1), (3, 1), (7, 2), (3, 4), (9, 2)}. © 1996 John Wiley & Sons, Inc.     

Almost resolvable Pk-decompositions of complete graphs

An almost P_k-factor of G is a P_k-factor of G - { v } for some vertex v. An almost resolvable P_k-decomposition of λK_n is a partition of the edges of λK_n into almost P_k-factors. We prove that necessary and sufficient conditions for the existence of an almost resolvable P_k-decomposition of λK_n are n ≡ 1 (mod k) and λnk/2 ≡ 0 (mod k ?1).     

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.     

Spectrum of resolvable directed quadruple systems

A t-(v, k, 1) directed design (or simply a t-(v, k, 1)DD) is a pair (S, ℐ), where S is a v-set and ℐ is a collection of k-tuples (called blocks) of S, such that every t-tuple of S belongs to a unique block. The t-(v, k, 1)DD is called resolvable if ℐ can be partitioned into some parallel classes, so that each parallel class is a partition of S. It is proved that a resolvable 3-(v, 4, 1)DD exists if and only if v = 0 (mod 4).     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

More large sets of resolvable MTS and DTS with even orders

In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.     

Support sizes of threefold resolvable triple systems

Let SS_R(v, 3) denote the set of all integer b* such that there exists a RTS(v, 3) with b* distinct triples. In this paper, we determine the set SS_R(v, 3) for v ≡ 3 (mod 6) and v ≥ 3 with only five undecided cases. We establish that SS_R(v, 3) = P(v, 3) for v ≡ 3 (mod 6), v ≥ 21 and v ≠ 33, 39 where P(v, 3) = {m_v, m_v + 4, m_v + 6, m_v + 7, …, 3m_v} and m_v, = v(v ? 1)/6. As a by‐product, we remove the last two undecided cases for the intersection numbers of Kirkman triple system of order 27, this improves the known result provided in [ 2 ]. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 275–289, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10037     

Minimum Resolvable Coverings of K v with Copies of K 4 ? e

Suppose K _v is the complete undirected graph with v vertices and K ₄ − e is the graph obtained from a complete graph K ₄ by removing one edge. Let (K ₄ − e)-MRC(v) denote a resolvable covering of K _v with copies of K ₄ − e with the minimum possible number n(v, K ₄ − e) of parallel classes. It is readily verified that n(v, K₄-e) 3 é2(v-1)/5 ù{n(v, K_4-e) \geq \lceil 2(v-1)/5 \rceil} . In this article, it is proved that there exists a (K ₄ − e)-MRC(v) with é2(v-1)/5 ù{\lceil 2(v-1)/5 \rceil} parallel classes if and only if v ≡ 0 (mod 4) with the possible exceptions of v = 108, 172, 228, 292, 296, 308, 412. In addition, the known results on the existence of maximum resolvable (K ₄ − e)-packings are also improved.     

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the v(v−1) off-diagonal cells can be resolved into v−1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of v−2 RILS(v)s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.     

Decomposition of complete multigraphs into crown graphs

Let λ K _v be the complete multigraph, G a finite simple graph. A G-design of λ K _v is denoted by GD(v,G,λ). The crown graph Q _n is obtained by joining single pendant edge to each vertex of an n-cycle. We give new constructions for Q _n -designs. Let v and λ be two positive integers. For n=4, 6, 8 and λ≥1, there exists a GD(v,Q _n ,λ) if and only if either (1) v>2n and λ v(v?1)≡0 (mod 4n), or (2) v=2n and λ≡0 (mod 4). Let n≥4 be even. Then (1) there exists a GD(2n,Q _n ,λ) if and only if λ≡0 (mod 4). (2) There exists a GD(2n+1,Q _n ,λ) when λ≡0 (mod 4).     

Small Uniformly Resolvable Designs for Block Sizes 3 and 4

A uniformly resolvable design (URD) is a resolvable design in which each parallel class contains blocks of only one block size k, such a class is denoted k‐pc and for a given k the number of k‐pcs is denoted r_k. In this paper, we consider the case of block sizes 3 and 4 (both existent). We use v to denote the number of points, in this case the necessary conditions imply that v ≡ 0 (mod 12). We prove that all admissible URDs with v < 200 points exist, with the possible exceptions of 13 values of r₄ over all permissible v. We obtain a URD({3, 4}; 276) with r₄ = 9 by direct construction use it to and complete the construction of all URD({3, 4}; v) with r₄ = 9. We prove that all admissible URDs for v ≡ 36 (mod 144), v ≡ 0 (mod 60), v ≡ 36 (mod 108), and v ≡ 24 (mod 48) exist, with a few possible exceptions. Recently, the existence of URDs for all admissible parameter sets with v ≡ 0 (mod 48) was settled, this together with the latter result gives the existence all admissible URDs for v ≡ 0 (mod 24), with a few possible exceptions.     

Existence of non‐resolvable Steiner triple systems

We consider two well‐known constructions for Steiner triple systems. The first construction is recursive and uses an STS(v) to produce a non‐resolvable STS(2v + 1), for v ≡ 1 (mod 6). The other construction is the Wilson construction that we specify to give a non‐resolvable STS(v), for v ≡ 3 (mod 6), v > 9. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 16–24, 2005.