Let (W,C) be an m-cycle system of order n and let Omega subset of W, \Omega \ = nu < n. We say that a handcuffed design (Omega, P) of order nu and block size s (2 less than or equal to s less than or equal to m - 1) is contained in (W, C) if for every p is an element of P there is an in-cycle c=(a(1), a(2),..., a(m)) is an element of C such that: (1) p=[a(k), a(k+1),..., a(k+s-1)] for some k is an element of (1,2,..., m) (i.e. the (s - 1)-path p occurs in the m-cycle c); and (2) a(k-1), a(k+s) is not an element of Omega. Note that in (1) and (2) all the indices are reduced to the range {1,...,m}{mod m}. For each n - 1 (mod 8) and for each s is an element of {2, 3} we determine all the integers nu such that there is a 4-cycle system of order n containing a handcuffed design of order nu and block size s. (C) 1999 Elsevier Science B.V. All rights reserved.

### Embedding handcuffed designs with block size 2 or 3 in 4-cycle systems

#### Abstract

Let (W,C) be an m-cycle system of order n and let Omega subset of W, \Omega \ = nu < n. We say that a handcuffed design (Omega, P) of order nu and block size s (2 less than or equal to s less than or equal to m - 1) is contained in (W, C) if for every p is an element of P there is an in-cycle c=(a(1), a(2),..., a(m)) is an element of C such that: (1) p=[a(k), a(k+1),..., a(k+s-1)] for some k is an element of (1,2,..., m) (i.e. the (s - 1)-path p occurs in the m-cycle c); and (2) a(k-1), a(k+s) is not an element of Omega. Note that in (1) and (2) all the indices are reduced to the range {1,...,m}{mod m}. For each n - 1 (mod 8) and for each s is an element of {2, 3} we determine all the integers nu such that there is a 4-cycle system of order n containing a handcuffed design of order nu and block size s. (C) 1999 Elsevier Science B.V. All rights reserved.
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1999
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11769/42530`
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