Group‐divisible designs with block size k having k + 1 groups,for k = 4, 5

We investigate the spectrum for k‐GDDs having k + 1 groups, where k = 4 or 5. We take advantage of new constructions introduced by R. S. Rees (Two new direct product‐type constructions for resolvable group‐divisible designs, J Combin Designs, 1 (1993), 15–26) to construct many new designs. For example, we show that a resolvable 4‐GDD of type g⁵ exists if and only if g ≡ 0 mod 12 and that a resolvable 5‐GDD of type g⁶ exists if and only if g ≡ 0 mod 20. We also show that a 4‐GDD of type g⁴m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 3 and 0 < m ≤ 3g/2, except possibly when (g,m) = (9,3) or (18,6), and that a 5‐GDD of type g⁵m¹ exists (with m > 0) if and only if g ≡ m ≡ 0 mod 4 and 0 < m ≤ 4g/3, with 32 possible exceptions. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 363–386, 2000     

The existence of uniform 5-GDDs

In this article, we construct group divisible designs (GDDs) with block size five, group-type g^u and index unity. The necessary condition for the existence of such a GDD is u ≷ 5, (u - 1)g ≡ 0 (mod 4) and u(u - 1)g² ≡ 0 (mod 20). It is shown that these necessary conditions are also sufficient, except possibly in a few cases. Additionally, a new construction to obtain GDDs using holey TDs is presented. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 275–299, 1997     

Holey Steiner pentagon systems

In this article, it is shown that the necessary conditions for the existence of a holey Steiner pentagon system (HSPS) of type hⁿ are also sufficient, except possibly for the following cases: (1) when n = 15, and h ≡ 1 or 5 (mod 6) where h ≢ 0 (mod 5), or h = 9; and (2) (h, n) ∈ {(6, 6), (6, 36), (15, 19), (15, 23), (15, 27), (30, 18), (30, 22), (30, 24)}. Moreover, the results of this article guarantee the analogous existence results for group divisible designs (GDDs) of type hⁿ with block-size k = 5 and index λ = 2. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 41–56, 1999     

The existence of large set of symmetric partitioned incomplete latin squares

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type g^u (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(g^u) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5).     

Kirkman packing and covering designs with spanned holes of size 2

A Kirkman holey packing (resp. covering) design, denoted by KHPD(g^u) (resp. KHCD(g^u)), is a resolvable (gu, 3, 1) packing (resp. covering) design of pairs with u disjoint holes of size g, which has the maximum (resp. minimum) possible number of parallel classes. Each parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3, and 4 when gu ≡ 2, 3, and 4 (mod 3) in turn. In this paper, the existence problem of a KHPD(2^u) and a KHCD(2^u) is solved with one possible exception of a KHPD(2⁸). © 2004 Wiley Periodicals, Inc.     

Incomplete group divisible designs with block size four and general index

In this paper, we investigate the existence of incomplete group divisible designs (IGDDs) with block size four, group-type (g, h) u and general index λ. The necessary conditions for the existence of such a design are that u ≥ 4, g ≥ 3h, λg(u 1) ≡ 0 (mod 3), λ(g h)(u 1) ≡ 0 (mod 3), and λu(u 1)(g 2 h 2 ) ≡ 0 (mod 12). These necessary conditions are shown to be sufficient for all λ≥ 2. The known existence result for λ = 1 is also improved.     

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 the doyen-wilson theorem for m-cycle systems

In this article we consider the embedding of m-cycle systems of order u in m-cycle systems of order v when m is odd. When u and v are 1 or m (mod 2m) we completely settle this problem, except possibly for the smallest such embedding in some cases when u ≡ v ≡ m (mod 2m). In particular, there are no exceptions if m ∈ {7,9}, so the generalization of the Doyen-Wilson Theorem is now settled for all odd m with m ≤ 9. © 1994 John Wiley & Sons, Inc.     

Block designs with large holes and α -resolvable BIBDs

In a (v, k, λ: w) incomplete block design (IBD) (or PBD [v, {k, w^*}. λ]), the relation v ≥ (k ? 1)w + 1 must hold. In the case of equality, the IBD is referred to as a block design with a large hole, and the existence of such a configuration is equivalent to the existence of a λ-resolvable BIBD(v ? w, k ? 1, λ). The existence of such configurations is investigated for the case of k = 5. Necessary and sufficient conditions are given for all v and λ ? 2 (mod 4), and for λ ≡ 2 mod 4 with 11 possible exceptions for v. © 1993 John Wiley & Sons, Inc.     

Group Divisible Designs with Block Size Four and Group Type gum1 with Minimum m

In this paper, we continue to investigate the spectrum for {4}-GDDs of type g^u m¹ with m as small as possible. We determine, for each admissible pair (g,u), the minimum values of m for which a {4}-GDD of type g^um¹ exists with four possible exceptions.Gennian Ge-Researcher supported by NSFC Grant 10471127.Alan C. H. Ling-Researcher supported by an ARO grant 19-01-1-0406 and a DOE grant.classification Primary 05B05     

Existence of GBRDs with block size 4 and BRDs with block size 5

Chaudhry et al. (J Stat Plann Inference 106:303–327, 2002) have examined the existence of BRD(v, 5, λ)s for \({\lambda \in \{4, 10, 20\}}\). In addition, Ge et al. (J Combin Math Combin Comput 46:3–45, 2003) have investigated the existence of \({{\rm GBRD}(v,4,\lambda; \mathbb{G}){\rm s}}\) when \({\mathbb{G}}\) is a direct product of cyclic groups of prime orders. For the first problem, necessary existence conditions are (i) v ≥ 5, (ii) λ(v ? 1) ≡ 0 (mod4), (iii) λ v(v ? 1) ≡ 0 (mod 40), (iv) λ ≡ 0 (mod 2). We show these are sufficient, except for \({v=5, \lambda \in \{4,10\}}\). For the second problem, we improve the known existence results. Five necessary existence conditions are (i) v ≥ 4, (ii) \({\lambda \equiv 0\;({\rm mod}\,|\mathbb{G}|)}\), (iii) λ(v ? 1) ≡ 0 (mod 3), (iv) λ v(v ? 1) ≡ 0 (mod 4), (v) if v = 4 and \({|\mathbb{G}| \equiv 2\;({\rm mod}\,4)}\) then λ ≡ 0 (mod 4). We show these conditions are sufficient, except for \({\lambda = |\mathbb{G}|, (v,|\mathbb{G}|) \in \{(4,3), (10,2), (5,6), (7,4)\}}\) and possibly for \({\lambda = |\mathbb{G}|, (v,|\mathbb{G}|) \in \{(10,2h), (5,6h), (7,4h)\}}\) with h ≡ 1 or 5 (mod 6), h > 1.     

On Group-Divisible Designs with Block Size Four and Group-Type g u m 1

We investigate the spectrum for {4}-GDDs of type g ^u m ¹. Wedetermine, for each even g, all values of m for which a {4}-GDD of typeg ^u m ¹ exists, for every fourth value of u. We similarlydetermine, for each odd g 11 or 17, all values of m for which a {4}-GDD of typeg ^u m ¹ exists, for every third value of u. Finally, weestablish, up to a finite number of values of u, the spectrum for {4}-GDDs of typeg ^u m ¹ where gu is even, g {11, 17}.     

Existence of HPMDs with block size five

In this article, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5 and type hⁿ, namely, n ≥ 5 and n(n - 1)hⁿ ≡ 0 (mod 5), is also sufficient, except possibly for a few cases. The results of this article guarantee the analogous existence results for group divisible designs (GDDs) of group-type hⁿ with block size k = 5 and having index λ = 4. Moreover, some more conclusive results for the existence of (v, 5, 1)-perfect Mendelsohn designs (PMDs) are also mentioned. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 257–273, 1997     

Super‐simple resolvable balanced incomplete block designs with block size 4 and index 3

The necessary conditions for the existence of a super‐simple resolvable balanced incomplete block design on v points with k = 4 and λ = 3, are that v ≥ 8 and v ≡ 0 mod 4. These conditions are shown to be sufficient except for v = 12. © 2003 Wiley Periodicals, Inc.     

The spectrum for large sets of disjoint mendelsohn triple systems with any index

The spectrum for LMTS(v,1) has been obtained by Kang and Lei (Bulletin of the ICA, 1993). In this article, firstly, we give the spectrum for LMTS(v,3). Furthermore, by the existence of LMTS(v,1) and LMTS(v,3), the spectrum for LMTS(v,λ) is completed, that is v ≡ 2 (mod λ), v ≥ λ + 2, if λ ? 0(mod 3) then v ? 2 (mod 3) and if λ = 1 then v ≠ 6. © 1994 John Wiley & Sons, Inc.     

Resolvable Mendelsohn triple systems with equal sized holes

An HMTS of type {n₁, n₂, ⋖, n_h} is a directed graph which can be decomposed into 3-circuits. If the 3-circuits can be partitioned into parallel classes, then the HMTS is called an RHMTS. In this article it is shown that the RHMTSs of type m^h exist when mh &equiv 0 (mod 3) and (m, h) &ne (1, 6), with the possible exception of h = 6 and , where M₁₇ = {m|m is divisible by a prime less than 17}. The existence of Mendelsohn frames, which is closely related to RHMTS, is also considered in this article. It is proved that a Mendelsohn frame of type t^u exists if and only if u ≥ 4 and t(u - 1) ≡ 0(mod 3) with 2 possible exceptions. © 1997 John Wiley & Sons, Inc. J Combin Designs 5:329–340, 1997     

On simple and supersimple transversal designs

In this paper, the existence of a transversal design TD_λ (4, g) is proved for all indices λ satisfying 2 ≤ λ ≤ g such that any two of its blocks intersect in at most two elements. Similar results are obtained for transversal designs without repeated blocks. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 311–320, 2000     

Improving two theorems of bose on difference families

In [2] R. C. Bose gives a sufficient condition for the existence of a (q, 5, 1) difference family in (GF(q), +)—where q ≡ 1 mod 20 is a prime power — with the property that every base block is a coset of the 5th roots of unity. Similarly he gives a sufficient condition for the existence of a (q, 4, 1) difference family in (GF(q, +)—where q ≡ 1 mod 12 is a prime power — with the property that every base block is the union of a coset of the 3rd roots of unity with zero. In this article we replace the mentioned sufficient conditions with necessary and sufficient ones. As a consequence, we obtain new infinite classes of simple difference families and hence new Steiner 2-designs with block sizes 4 and 5. In particular, we get a (p^4α, 5, 1)-DF for any odd prime p ≡ 2, 3 (mod 5), and a (p^2α, 4, 1)-DF for any odd prime p ≡ 2 (mod 3). © 1995 John Wiley & Sons, Inc.     

Existence of directed GDDs with block size five

In this article, we construct directed group divisible designs (DGDDs) with block size five, group-type hⁿ, and index unity. The necessary conditions for the existence of such a DGDD are n ≥ 5, (n − 1)h ≡ 0 (mod 2) and n(n − 1)h² ≡ 0 (mod 10). It is shown that these necessary conditions are also sufficient, except possibly for n = 15 where h ≡ 1 or 5 (mod 6) and h ≢ 0 (mod 5), or (n, h) = (15, 9). © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 389–402, 1998     

Kirkman covering designs with even-sized holes

A Kirkman holey covering design, denoted by KHCD(g^u), is a resolvable group-divisible covering design of type g^u. Each of its parallel class contains one block of size δ, while other blocks have size 3. Here δ is equal to 2, 3 and 4 when gu≡2, 3 and 4 (mod 3) in turn. In this paper, we study the existence problem of a KHCD(g^u) which has minimum possible number of parallel classes, and give a solution for most values of even g and u.