3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

Coloring BIBDs with block size 4

It is shown that for each integer m ≥ 1 there exists a lower bound, v_m, with the property that for all v ≥ v_m with v ≡ 1, 4 (mod 12) there exists an m-chromatic S(2, 4, v) design. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 403–409, 1998     

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     

Completing the spectrum for LGDD (mv)

In this article, we construct LGDD(m^v) for v ≡ 2 (mod 6) and m ≡ 0 (mod 6) by constructing the missing essential values 6^v, v = 8, 14, 26, 50. Thus an LGDD(m^v) exists if and only if v(v - 1)m² ≡ 0 (mod 6), (v - 1)m ≡ 0 (mod 2) and (m, v) ≠ (1, 7). © 1997 John Wiley & Sons, Inc.     

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.     

On the minimum number of edge-disjoint complete m-partite subgraphs into which Kn can be decomposed

Let α_m(n) denote the minimum number of edge-disjoint complete m-partite subgraphs into which K_n can be decomposed. In [2] the author proved that when m ≥ 3, if (i) n ≡ m and n ≡ m (mod m ?1), or (ii) b ∈ [2, m ?1], n ≥ b(m ?1) + m ? (b ?1), and n ≡ b(m ?1) + m ? (b ?1) (mod m? 1), then α_m(n) = ?(n + m ?3)/(m ?1)? (= ?(n ?1)/(m ?1)?), and that for every integer n, if K_n has an edge-disjoint complete m-partite subgraph decomposition, then α_m(n) ≥ ?(n? 1)/(m? 1)?. In this paper we generally discuss the question as to which integers n's satisfy (or do not) α_m(n) = ?(n ?1)/(m ?1)?. Here we also study the methods to find these integers; the methods are themselves interesting. Our main results are Theorem 2.11, 2.12, and 2.16. Besides, Theorem 2.4 and 2.6 are interesting results too. © 1993 John Wiley & Sons, Inc.     

Further results on large sets of disjoint group‐divisible designs with block size three and type 2n41

Large sets of disjoint group‐divisible designs with block size three and type 2ⁿ4¹ were first studied by Schellenberg and Stinson because of their connection with perfect threshold schemes. It is known that such large sets can exist only for n ≡0 (mod 3) and do exist for all odd n ≡ (mod 3) and for even n=24m, where m odd ≥ 1. In this paper, we show that such large sets exist also for n=2^k(3m), where m odd≥ 1 and k≥ 5. To accomplish this, we present two quadrupling constructions and two tripling constructions for a special large set called ^*LS(2ⁿ). © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 24–35, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10032     

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.     

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.     

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.     

A bound for Wilson's theorem (III)

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

Simple and indecomposable twofold cyclic triple systems from Skolem sequences

We construct simple indecomposable twofold cyclic triple systems TS₂(v) for all v ≡ 0, 1, 3, 4, 7, and 9(mod 12), where v = 4 or v ≥ 12, using Skolem‐type sequences. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 402–410, 2000     

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.     

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.     

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.     

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.     

On large sets of almost Hamilton cycle decompositions

A large set of CS(v, k, λ), k‐cycle system of order v with index λ, is a partition of all k‐cycles of K_v into CS(v, k, λ)s, denoted by LCS(v, k, λ). A (v ? 1)‐cycle is called almost Hamilton. The completion of the existence spectrum for LCS(v, v ? 1, λ) only depends on one case: all v ≥ 4 for λ = 2. In this article, it is shown that there exists an LCS(v, v ? 1,2) for any v ≡ 0,1 (mod 4) except v = 5, and for v = 6,7,10,11. © 2006 Wiley Periodicals, Inc. J Combin Designs 16: 53–69, 2008     

Some results on (v,{5,w*})-pbds

In this article, we construct pairwise balanced designs (PBDs) on v points having blocks of size five, except for one block of size w ? {17,21,25,29,33}. A necessary condition for the existence of such a PBD is v ? 4w + 1 and (1) v ≡ 1 or 5 (mod 20) for w = 21, 25; (2) v ≡ 9 or 17 (mod 20) for w = 17,29; (3) v ≡ 13 (mod 20) for w = 33. We show that these necessary conditions are sufficient with at most 25 possible exceptions of (v,w). We also show that a BIBD B(5, 1; w) can be embedded in some B(5, 1; v) whenever v ≡ w ≡ 1 or 5 (mod 20) and v ? 5w ? 4, except possibly for (v, w) = (425, 65). © 1995 John Wiley & Sons, Inc.     

A bound for wilson's theorem (I)

In 1975, Richard M. Wilson proved: Given any positive integers k ? 3 and λ, there exists a constant v⁰ = v⁰(k, λ) such that v ? B(k,λ) for every integer v ? v⁰ that satisfies λ(v ? 1) ≡ 0(mod k ? 1) and λv(v ? 1) ≡ 0[mod k(k ? 1)]. The proof given by Wilson does not provide an explicit value of v⁰. We try to find such a value v⁰(k, λ). In this article we consider the case λ = 1 and v ≡ 1[mod k(k ? 1)]. We show that: if k ? 3 and v = 1[mod k(k ? 1)] where v > k^kk5, then a B(v,k, 1) exists. © 1995 John Wiley & Sons, Inc.     

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.