Extended skolem sequences

A k-extended Skolem sequence of order n is an integer sequence (s₁, s₂,…, s_2n+1) in which s_k = 0 and for each j ? {1,…,n}, there exists a unique i ? {1,…, 2n} such that s_i = s_i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.     

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).     

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.     

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.     

Note on group distance magic complete bipartite graphs

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ? from V to an Abelian group Γ of order n such that the weight $w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ? _p -distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K _m,n is a group distance magic graph if and only if n + m ? 2 (mod 4).     

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     

More cyclic triplewhist tournaments

We extend the set of values of n for which it is known that a Z-cyclic triple whist tournament for 4n players exists by proving that if there exists such a tournament for q + 1 players, where q ≡ 3 (mod 4) is prime, then there exists such a tournament for qp^a1₁…p^an_n + 1 players, whenever the p_i are primes ≡ 5 (mod 8). © 1995 John Wiley & Sons, Inc.     

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     

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).     

Fixed Points of Automorphisms of Graphs with 1 – Factorizations

We consider sets of fixed points of finite simple undirected connected graphs with 1 – factorizations. The maximum number of fixed points of complete graphs K_2n (n > 2) is n if n ≡ 0 mod 4, n — 1 if n ≡ 3 mod 4 or n ≡ 5 mod 12, n — 2 if n ≡ 2 mod 4, n — 3 if n ≡ 1 mod 4 and n ≢ 5 (mod 12). The maximum number of fixed points of 1 – factorizations of (non – complete) graphs with 2n vertices is less than or equal to n. If n is a prime number, then there are graphs with 2n vertices whose 1 – factorizations have automorphisms with n fixed points. Moreover, a result on the structure of a group of fixed – point – free automorphisms is presented.     

Path coverings with paths

In this paper, we give two new proofs of a result of Heinrich, Langdeau and Verrall that provide necessary and sufficient conditions for the existence of a set S of 3‐paths in K_n having the property that each 2‐path in K_n lies in exactly one path in S. These are then used to consider the case n ≡ 3 (mod 4) when no such exact covering is possible, and to solve the problem of covering (k−1)‐paths with k‐paths for all k ≥ 3. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 156–167, 2001     

On tree factorizations of Kn

In this paper we exhibit a class of trees with the property that if T_k is a tree on k vertices that belongs to this class, then necessary and sufficient conditions for K_n to have a T_k-factorization are simply n = 0 (mod k) and n = 1 (mod 2(k - 1)). (This class is large and in particular contains all caterpillars with an odd number of vertices.) As a corollary we obtain necessary and sufficient conditions for the existence of a K_1,k-1-factorization of K_n, which previously had only been known to be asymptotically sufficient. © 1993 John Wiley & Sons, Inc.     

Doyen–Wilson theorem for nested Steiner triple systems

The obvious necessary conditions for the existence of a nested Steiner triple system of order v containing a nested subsystem of order w are v ≥ 3w + 4 and v ≡ w ≡ 1 (mod 6). We show that these conditions are also sufficient. © 2004 Wiley Periodicals, Inc.     

Large sets of 3-designs from psl(2, q), with block sizes 4 and 5

We determine the distribution of 3?(q + 1,k,λ) designs, with k ? {4,5}, among the orbits of k-element subsets under the action of PSL(2,q), for q ? 3 (mod 4), on the projective line. As a consequence, we give necessary and sufficient conditions for the existence of a uniformly-PSL(2,q) large set of 3?(q + 1,k,λ) designs, with k ? {4,5} and q ≡ 3 (mod 4). © 1995 John Wiley & Sons, Inc.     

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     

An infinite family of Hadamard matrices of Williamson type

In this paper a new proof is given of the following theorem of Turyn: Let q = 2n ? 1 be a prime power ≡1 (mod 4); then there exists an Hadamard matrix of order 4n that is of the Williamson type.     

Existence of 1-Rotational <Emphasis Type="Italic">k</Emphasis>-Cycle Systems of the Complete Graph

We explicitly solve the existence problem for 1-rotational k-cycle systems of the complete graph K_v with v1 or k (mod 2k). For v1 (mod 2k) we have existence if and only if k is an odd composite number. For any odd k and vk (mod 2k), (except k3 and v15, 21 (mod 24)) a 1-rotational k-cycle system of K_v exists.Final version received: June 18, 2003     

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.     

Existence of (q,k, 1) difference families with q a prime power and k = 4, 5

The existence of a (q, k, 1) difference family in GF(q) has been completely solved for k = 3. For k = 4, 5 partial results have been given by Bose, Wilson, and Buratti. In this article, we continue the investigation and show that the necessary condition for the existence of a (q, k, 1) difference family in GF(q), i.e., q ≡ 1 (mod k(k − 1)) is also sufficient for k = 4, 5. For general k, Wilson's bound shows that a (q, k, 1) difference family in GF(q) exists whenever q ≡ 1 (mod k(k − 1)) and q > [k(k − 1)/2]^k(k−1). An improved bound on q is also presented. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 21–30, 1999