On the 3BD‐closed set B3({4,5,6})

Let B₃(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B₃(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if v ∈ B₃ ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B₃{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B₃({4,5,6}) by Hanani and that B₃({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B₃({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B₃({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.     

Z-cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5

When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all v ∈ T = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all v ∈ T there exists a Z-cyclic TWh(vⁿ) for all n ≥ 1, and (2) if v_i ∈ T, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?_i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.     

Triangles in 2-factorizations

The triangle-spectrum for 2-factorizations of the complete graph K_v is the set of all numbers δ such that there exists a 2-factorization of K_v in which the total number of triangles equals δ. By applying mainly design-theoretic methods, we determine the triangle spectrum for all v ≡ 1 or 3 (mod 6), v ≥ 43, as well as for v = 7, 9, 13, 15, 21, and 27. For orders v = 19, 25, 31, 33, 37, 39, we leave only a total of 11 values undecided. To determine the triangle-spectrum for v ≡ 5 (mod 6) remains an open problem. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 83–94, 1997     

Cyclic k‐cycle systems of order 2kn + k: A solution of the last open cases

We exhibit cyclic (K_v, C_k)‐designs with v > k, v ≡ k (mod 2k), for k an odd prime power but not a prime, and for k = 15. Such values were the only ones not to be analyzed yet, under the hypothesis v ≡ k (mod 2k). Our construction avails of Rosa sequences and approximates the Hamiltonian case (v = k), which is known to admit no cyclic design with the same values of k. As a particular consequence, we settle the existence question for cyclic (K_v, C_k)‐designs with k a prime power. © 2004 Wiley Periodicals, Inc. J Combin Designs 12: 299–310, 2004.     

Q5‐factorization of λKn

Q_k is the simple graph whose vertices are the k‐tuples with entries in {0, 1} and edges are the pairs of k‐tuples that differ in exactly one position. In this paper, we proved that there exists a Q₅‐factorization of λK_n if and only if (a) n ≡ 0(mod 32) if λ ≡ 0(mod 5) and (b) n ≡ 96(mod 160) if λ ? 0(mod 5).     

Some infinite families of large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection {(X, B_i) : 1 ≤ i ≤ v ? 2}, where every (X,B_i) is a KTS(v) and all B_i form a partition of all triples on X. Many researchers have studied the existence of LKTS(v) for a long time. In [13], the author introduced a concept—large set of generalized Kirkman systems (LGKS), which plays an important role in the discussion of LKTS. In this article, we give a new construction for LGKS and obtain some new results of LKTS, that is, there exists an LKTS(6u + 3) for u = qⁿ, where n ≥ 1, q ≡ 7 (mod 12) and q is a prime power. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 202–212, 2008     

The fine triangle intersections for maximum kite packings

In this paper, the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v) = {(s,t) : a pair of maximum kite packings of order v intersecting in s blocks and s+t triangles}. Let Adm(v) = {(s, t) : s + t ≤ bv , s,t are non-negative integers}, where b v = v(v 1)/8 . It is established that Fin(v) = Adm(v)\{(bv-1, 0), (bv-1,1)} for any integer v ≡ 0, 1 (mod 8) and v ≥ 8; Fin(v) = Adm(v) for any integer v ≡ 2, 3, 4, 5, 6, 7 (mod 8) and v ≥ 4.     

Resolvable packings of Kv with K2 × Kc's

The study of resolvable packings of K_v with K_r × K_c's is motivated by the use of DNA library screening. We call such a packing a (v, K_r × K_c, 1)‐RP. As usual, a (v, K_r × K_c, 1)‐RP with the largest possible number of parallel classes (or, equivalently, the largest possible number of blocks) is called optimal. The resolvability implies v ≡ 0 (mod rc). Let ρ be the number of parallel classes of a (v, K_r × K_c, 1)‐RP. Then we have ρ ≤ ?(v‐1)/(r + c ? 2)?. In this article, we present a number of constructive methods to obtain optimal (v, K₂ × K_c, 1)‐RPs meeting the aforementioned bound and establish some existence results. In particular, we show that an optimal (v, K₂ × K₃, 1)‐RP meeting the bound exists if and only if v ≡ 0 (mod 6). © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 177–189, 2009     

On the decomposition of λK into regular tournaments of order 5

An RTD[5,λ; v] is a decomposition of the complete symmetric directed multigraph, denoted by λK, into regular tournaments of order 5. In this article we show that an RTD[5,λ; v] exists if and only if (v?1)λ ≡ 0 (mod 2) and v(v?1)λ ≡ 0 (mod 10), except for the impossible case (v,λ) = (15,1). Furthermore, we show that for each v ≡ 1,5 (mod 20), v ≠ 5, there exists a B[5,2; v] which is not RT₅-directable. © 1994 John Wiley & Sons, Inc.     

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.     

Directed B(K, 1; v) with K = {4, 5} and {4, 6} related to deletion/insertion‐correcting codes

Motivated by the construction of t‐deletion/insertion‐correcting codes, we consider the existence of directed PBDs with block sizes from K = {4, 5} and {4, 6}. The spectra of such designs are determined completely in this paper. For any integer {υ ≥ 4, a DB({4,5} ,1; υ) exists if and only if υ∉{6, 8, 9, 12, 14}, and a DB({4, 6}, 1; υ) exists if and only if υ ≡ 0,1 mod 3 and υ∉{9,15}. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 147–156, 2001     

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.     

Bounds on a generalized domination parameter

Abstract

If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be K_n-adjacent vertices of G if there is a subgraph of G, isomorphic to K_n , containing u and v. For n ≥ 2, a K_n- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is K_n-adjacent to a vertex of D. The K_n-domination number γK_n (G) of G is the minimum cardinality among the K_n-dominating sets of vertices of G. It is shown that, for n ε {3,4}, if G is a graph of order p with no K_n-isolated vertex, then γK_n (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.     

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.     

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     

A bound for Wilson's theorem (II)

In this article we prove the following theorem. For any k ≥ 3, let c(k, 1) = exp{exp{k^k2}}. If v(v − 1) ≡ 0 (mod k(k −1)) and v − 1 ≡ 0 (mod k−1) and v > c(k, 1), then a B(v,k, 1) exists. © 1996 John Wiley & Sons, Inc.     

On partitions of {1,…,2m + 1}\{k} into differences d,…,d + m − 1: Extended Langford sequences of large defect

It is shown that for m = 2d ? 1, 2d, 2d + 1, and d ≥ 1, the set {1, 2,…, 2m + 2}, ? {2,k} can be partitioned into differences d,d + 1,…,d + m ? 1 whenever (m,k) ≡ (0,0), (1,d + 1), (2, 1), (3,d) (mod (4,2)) and (d,m,k) ≠ (1,1,3), (2,3,7) (where (x,y) ≡ (u,ν) mod (m,n) iff x ≡ u (mod m) and y ≡ ν (mod n)). It is also shown that if m ≥ 2d ? 1 and m ? [2d + 2, 8d ? 5], then the set {1, 2, …, 2m + 1} ? {k} can be partitioned into differences d,d + 1,…,d + m ? 1 whenever (m,k) ≡ (0, 1), (1,d), (2,0), (3,d + 1) mod (4,2). Finally, for d = 4 we obtain a complete result for when {1,…,2m + 1} ? {k} can be partitioned into differences 4,5,…,m + 3. © 2004 Wiley Periodicals, Inc.     

A new infinite class of S(3, {4, 5, 7}, v)

A t‐wise balanced design ( at BD) of order v and block sizes from K , denoted by S ( t , K , v ), is a pair ( X , ??), where X is a v ‐element set and ?? is a set of subsets of X , called blocks , with the property that | B |∈ K for any B ∈?? and every t ‐element subset of X is contained in a unique block. In this article, we shall show that there is an S ( 3 , { 4 , 5 , 7 }, v ) for any positive integer v ≡ 7 ( mod12 ) with v ≠ 19 . Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:68–80, 2012     

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.     

Pairwise balanced designs whose block size set contains seven and thirteen

In this paper, we investigate the PBD‐closure of sets K with {7,13} ? K ? {7,13,19,25,31,37,43}. In particular, we show that ν ≡ 1 mod 6, ν ≥ 98689 implies ν ? B({7,13}). As an intermediate result, many new 13‐GDDs of type 13^q and resolvable BIBD with block size 6 or 12 are also constructed. Furthermore, we show some elements to be not essential in a Wilson basis for the PBD‐closed set {ν: ν ≡ 1 mod 6, ν ≥ 7}. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 283–314, 2007