Embedding path designs in 4-cycle systems

Let (W, C) be an m-cycle system of order n and let Ω ⊂ W , |Ω| = v < n. We say that a path design (Ω,P) of order v and block size s (2 ≤ s ≤ m− 1) is embedded in (W, C) if for every p ∈ P there is an m-cycle c = (a1, a2, . . . , am) ∈ C such that: (1) p = [ak, ak+1, . . . , ak+s−1] for some k ∈ {1, 2, . . . ,m} (i.e. the (s− 1)-path p occurs in the m-cycle c); and (2) ak−1, ak+s ∈ Ω. 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 ∈ {2, 3}, the spectrum of all the integers v such that there is a handcuffed design of order v and block size s embedded in a 4-cycle system of order n is determined in [5]. In this paper we want to complete the case m = 4 by determining the set of all the integers v such that there is a path design of order v and block size 3 embedded in a 4-cycle system of order n. AMS classification: 05B05.