Frank E. Bennett

Perfect Mendelsohn designs with block size six

Abstract A necessary condition for the existence of a (v,k,λ)-perfect Mendelsohn design is λv(v−1)≡0 (mod k). For k=6 and λ=1, this condition gives v≡0,1,3 or 4 (mod 6). Miao and Zhu have investigated the cases v≡0,1 (mod 6). This paper provides several improvements on their results and also investigates the cases v≡3,4 (mod 6). For v≡1 (mod 6) we solve the problem completely; for v≡0,3,4 (mod 6), the largest unknown cases are for v=198,657,148, respectively.

Constructions of perfect Mendelsohn designs

Abstract Let n and k be positive integers. An (n, k, 1)-Mendelsohn design is an ordered pair (V, C ) where V is the vertex set of Dn, the complete directed graph on n vertices, and C is a set of directed cycles (called blocks) of length k which form an arc-disjoint decomposition of Dn. An (n, k, 1)-Mendelsohn design is called a perfect design and denoted briefly by (n, k, 1)-PMD if for any r, 1⩽r⩽k-1, and for each (x, y) ϵ V×V there is exactly one cycle c∈ C in which the (directed) distance along c from x to y is r. A necessary condition for the existence of an (n, k, 1)-PMD is n(n-1)≡0(modk). In this paper we shall describe some new techniques used in the construction of PMDs, including constructions of the product type. As an application, we show that the necessary condition for the existence of an (n, 5, 1)-PMD is also sufficient, except for n=6 and with at most 21 possible exceptions of n of which 286 is the largest.

On minimum matrix representation of closure operations

Abstract Let b be a column of an m×n matrix M and A a set of its columns. We say that A implies b if and only if M contains no two rows equal in A but different in b. If L M(A) denotes the set of columns implied by A, then L M(A) is a closure operation and we say that M represents this closure operation. Let s( L ) denote the minimum number of rows of the matrices representing a given closure operation L . The k-uniform closure operation on an n-element groundset X is defined by L k n (A)= X, if |A|≥k, A, if |A| . It is known that the closure operation L n k can be represent by some matrix, that is, there is an m×n matrix M such that L M = L n k . In this paper we shall verify two conjectures of Demetrovics et al. with regards to the determination of s( L n 3 ) . In particular, we prove that s( L n 3 )=n for all n ≥ 7, with the possible exception of n = 8.

Quintessential pairwise balanced designs

The existence of pairwise balanced designs with block sizes from a set K is studied. The spectrum of orders for which such PBDs exist is determined when {5}⊂K⊆{5,6,7,8,9}, with relatively few possible exceptions in each case.

Super-simple Steiner pentagon systems

A Steiner pentagon system of order v(SPS(v)) is said to be super-simple if its underlying (v,5,2)-BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. In this paper, it is shown that the necessary condition for the existence of a super-simple SPS(v); namely, v>=5 and v=1 or 5(mod10) is sufficient, except for v=5, 15 and possibly for v=25. In the process, we also improve an earlier result for the spectrum of super-simple (v,5,2)-BIBDs, removing all the possible exceptions. We also give some new examples of Steiner pentagon packing and covering designs (SPPDs and SPCDs).

Existence of (v; {5 ;w ∗ }; 1)-PBDs

Abstract In this paper, we investigate the existence of pairwise balanced designs on v points having blocks of size five, with a distinguished block of size w, briefly (v,{5,w ∗ },1) -PBDs. The necessary conditions for the existence of a (v,{5,w ∗ },1) -PBD with a distinguished block of size w are that v⩾4w+1, v≡w≡1 ( mod 4) and either v≡w ( mod 20) or v+w≡6 ( mod 20) . Previously, w⩽33 has been studied, and the necessary conditions are known to be sufficient for w=1, 5, 13 and 21, with 8 possible exceptions when w⩽33. In this article, we eliminate 3 of these possible exceptions, showing sufficiency for w=25 and 33. Our main objective is the study of 37⩽w⩽97, where we establish sufficiency for w=73, 81, 85 and 93, with 67 possible exceptions with 37⩽w⩽97. For w≡13 ( mod 20) , we show that the necessary existence conditions are sufficient except possibly for w=53,133,293 and 453. For w≡1,5 ( mod 20) , we show the necessary existence conditions are sufficient for w⩾1281,1505, and for w≡9,17 ( mod 20) , we show that w⩾2029,2477 is sufficient with one possible exceptional series, namely v=4w+9 when w≡17 ( mod 20) . We know of no example where v=4w+9. In this article, we also study the 4-RBIBD embedding problem for small subdesigns (up to 52 points) and update some results of Bennett et al. on PBDs containing a 5-line. As an application of our results for w=33 and 97, we establish the smallest number of blocks in a pair covering design with k=5 when v≡1 ( mod 4) with 37 open cases, the largest being for v=489; hitherto, there were 104 open cases, the largest being v=2249.

The Existence of Four HMOLS with Equal Sized Holes

In this paper, we present several new constructions for k holey mutually orthogonal Latin squares (HMOLS) of type gn. We concentrate mainly on k=4; here, for all but two values of n, namely 6 and 15, only a finite number of unsolved cases remain. Some new sets of 5 and 6 HMOLS are also given, in particular 5 HMOLS(2q) for q≥63 or q an odd prime power between 6 and 62, plus 6 HMOLS(4q)for q an odd prime power between 8 and 60.

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.

The spectrum of HSOLSSOM( h n ) where h is even

Abstract In this paper, we describe a generalized product construction and a construction using Steiner pentagon systems to obtain holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOM). We investigate the existence of HSOLSSOM( h n ) for even h . We first improve the known result for h = 2 and show that a HSOLSSOM(2 n ) exists for all n ⩾ 5 except possibly for n ∈ E = {8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 28, 32}. As a consequence, we then establish that for h  2 (mod 4) and h ⩾ 10 a HSOLSSOM( h n ) exists for all n ⩾ 5 except possibly for n ϵ E . More conclusively, we show that for h  0 (mod 4), a HSOLSSOM( h n ) exists if and only if n ⩾ 5. We are also able to apply our results to construct a unipotent SOLSSOM(62), the existence of which was previously unknown.

Existence of three HMOLS of types h n and 2 n 3 1

Abstract In this paper, we construct 23 new 3 HMOLS of type h n . We also investigate the existence of 3 HMOLS of type 2 n 3 1 and show that the necessary condition n ⩾ 6 is sufficient for such designs to exist except possibly for 18 values of n , of which n = 31 is the largest. As an application, some improvements for the existence of perfect Mendelsohn designs with block size five are also mentioned.