Configurations in 4-cycle systems

Bryant, Darryn; Grannell, Mike; Griggs, Terry and Macaj, Martin (2004). Configurations in 4-cycle systems. Graphs and Combinatorics, 20(2) pp. 161–179.

DOI: https://doi.org/10.1007/s00373-004-0553-4

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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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