Abstract
LetX be a set ofv + 1 elements, wherev ≡ 0 or 1 (mod 3). If two copies of the collection of\(\left( {\begin{array}{*{20}c} {v + 1} \\ 3 \\ \end{array} } \right)\) triples chosen fromX can be partitioned intov + 1 mutually disjoint two-fold triple systems, each based on a differentv-subset ofX, we say they form an overlarge set of two-fold triple systems, denoted byOS(TTS(v)). In this paper, we give the first construction of anOS(TTS(10)). We then investigate the properties of the uniqueOS(TTS(6)) and obtain:
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(i)
A partition of the set of 84 distinctTTS(6) based onX = {1, 2,..., 7} into 12 parallel classes, that is, into 12OS(TTS(6)) each containing sevenTTS(6);
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(ii)
A resolution of the set of 1008 distinctOS(TTS(6)) based onX into 84 parallel classes;
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(iii)
Simple constructions of several strongly-regular graphs, including both the Hoffman-Singleton and Higman-Sims graphs, which arise from the relation between the family of 84 distinctTTS(6) and the family of 30 distinctSTS(7), all based onX.
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Supported by NSERC grant A8651.
Supported by ARC grant A49130102 and an Australian Senior Research Fellowship
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Mathon, R., Street, A.P. Partitions of sets of two-fold triple systems, and their relation to some strongly regular graphs. Graphs and Combinatorics 11, 347–366 (1995). https://doi.org/10.1007/BF01787815
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DOI: https://doi.org/10.1007/BF01787815