Abstract.
A 4-cycle system of order n, denoted by 4CS(n), exists if and only if n≡1 (mod 8). There are four configurations which can be formed by two 4-cycles in a 4CS(n). Formulas connecting the number of occurrences of each such configuration in a 4CS(n) are given. The number of occurrences of each configuration is determined completely by the number d of occurrences of the configuration D consisting of two 4-cycles sharing a common diagonal. It is shown that for every n≡1 (mod 8) there exists a 4CS(n) which avoids the configuration D, i.e. for which d=0. The exact upper bound for d in a 4CS(n) is also determined.
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Acknowledgments A substantial part of the work forming this paper was done while the first author was visiting the Department of Pure Mathematics of The Open University at Milton Keynes; he thanks the Department for hospitality and the UK EPSRC for financial support (grant number GR/R78282/01). The first author also gratefully acknowledges the support of the Australian Research Council. The fourth author is supported by the VEGA grant 1/0261/03. All the authors thank the referees for their helpful comments.
Final version received: November 11, 2003
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Bryant, D., Grannell, M., Griggs, T. et al. Configurations in 4-Cycle Systems. Graphs and Combinatorics 20, 161–179 (2004). https://doi.org/10.1007/s00373-004-0553-4
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DOI: https://doi.org/10.1007/s00373-004-0553-4