R.J.R. Abel
University of New South Wales
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Featured researches published by R.J.R. Abel.
Journal of Statistical Planning and Inference | 2000
R.J.R. Abel; Frank E. Bennett; Hantao Zhang
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.
Discrete Applied Mathematics | 2008
R.J.R. Abel; Frank E. Bennett
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).
Journal of Combinatorial Designs | 1999
R.J.R. Abel; Frank E. Bennett; Hantao Zhang
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.
Designs, Codes and Cryptography | 1997
R.J.R. Abel; Charles J. Colbourn; Jianxing Yin; Hantao Zhang
The basic necessary condition for the existence of a TD(5, λ; v)-TD(5, λ; u), namely v ≥ 4u, is shown to be sufficient for any λ ≥ 1, except when (v, u) = (6, 1) and λ = 1, and possibly when (v, u) = (10, 1) or (52, 6) and λ = 1. For the case λ = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.
Discrete Mathematics | 1998
R.J.R. Abel; F.E. Bennett
Abstract Necessary conditions for existence of a ( v,k,λ ) perfect Mendelsohn design (or PMD) are v ⩾ k and λv ( v − 1) ≡ 0 mod k . When k = 7, this condition is satisfied if v ≡ 0 or 1 mod 7 and v ⩾ 7 when λ ≢ 0 mod 7 and for all v ⩾ 7 when λ ≡ 0 mod 7. Bennett, Yin and Zhu have investigated the existence problem for k = 7, λ = 1 and λ even; here we provide several improvements on their results and also investigate the situation for λ odd. We reduce the total number of unknown ( v ,7, λ )-PMDs to 36,31 for λ = 1 and 5 for λ > 1. In particular, v = 294 is the largest unknown case for λ = 1, and the only unknown cases for λ > 1 are for v = 42, λ ∈ [2,3,5,9] and v = 18, λ = 7.
Discrete Applied Mathematics | 2004
R. Julian; R.J.R. Abel; Stephanie Costa; Norman J. Finizio
In this paper a new specialization of whist tournament is introduced, namely a directed-ordered whist tournament. It is established that directed-ordered whist tournaments do not exist when the number of players, v, equals 4n and that directed -ordered whist tournaments exist for all v=4n+1. Several new (v, 5, 1) difference families are given and are combined with a construction of Buratti and Zuanni to produce Z-cyclic directed-ordered whist tournaments. Infinite families of Z-cyclic directed-ordered whist tournaments are obtained by applying the product theorems of Anderson et al. to these latter designs together with the classic whist construction of Baker which is shown to produce directed-ordered whist designs. In addition many new examples of Z-cyclic directed whist tournaments and ordered whist tournaments are given.
Discrete Mathematics | 2002
R.J.R. Abel; Frank E. Bennett; Gennian Ge; L. Zhu
Abstract A Steiner k -cycle system of order v is a pair (X,C) , where C is a collection of k -cycles of K v based on a v -set X such that for any integer r, 1⩽r⩽k/2 , and for any two distinct vertices x and y of X there exists in C a unique k -cycle along which the distance between x and y is r . Steiner k -cycle systems are useful in constructing authentication perpendicular arrays and authentication and secrecy codes. In this paper, we show that the necessary condition for the existence of Steiner seven-cycle systems, v≡1 or 7 ( mod 14) , is also sufficient if v>861 . We also show that there are at most 21 unknown orders below this bound. The result is mainly based on generalized constructions for two holey self-orthogonal Latin squares with symmetric orthogonal mates (2 HSOLSSOM) and some direct constructions. As an application, we shall update the known result on the existence of perfect Mendelsohn designs with block size 7 .
Discrete Mathematics | 2001
R.J.R. Abel; Frank E. Bennett; Hantao Zhang; L. Zhu
Abstract Let K n denote the complete undirected graph on n vertices. A Steiner pentagon covering design (SPCD) of order n is a pair (K n , B ) , where B is a collection of c ( n )=⌈ n /5⌈ n −1/2⌉⌉ pentagons from K n such that any two vertices are joined by a path of length 1 in at least one pentagon of B , and also by a path of length 2 in at least one pentagon of B . The existence of SPCDs is investigated. The main approach is to use certain types of holey Steiner pentagon systems. For n even, the existence of SPCDs is established with a few possible exceptions. For n odd, new SPCDs are found which improve an earlier known result. In addition, new results are also found for Steiner pentagon packing designs.
Discrete Mathematics | 2001
R. Julian; R.J.R. Abel; Malcolm Greig; Ying Miao; L. Zhu
Abstract A necessary condition for the existence of a resolvable balanced incomplete block design on v points with k=7 and λ=6 is that v≡0 mod 7 . This condition is shown to be sufficient for v>462 , with 18 possible exceptions plus one known exception (v=14) below this value. Also considered are the existence of (v,7,6) -NRDs, (7,6)-frames of type 7 t and (7,6)-resolvable GDDs of type 7 t .
Discrete Applied Mathematics | 2008
R.J.R. Abel; Frank E. Bennett; Gennian Ge
A directed triplewhist tournament on v players, briefly DTWh(v), is said to have the three person property if no two games in the tournament have three common players. We briefly denote such a design as a 3PDTWh(v). In this paper, we show that a 3PDTWh(v) exists whenever v>17 and v=1(mod4) with few possible exceptions.