Embedding Path Designs in 4-Cycle Systems

Let (, )bean -cycle system of order and let . We say that a path design (Ω, ) of order and block size (2 ≤ ≤ — 1) in (, ) if for every ∈ there is an -cycle such that: (1) for some (i.e. the ( – 1)-path occurs in the -cycle ); and (2) . Note that in (1) and (2) all the indices are reduced to the range {} (mod ).For each ≡ 1 (mod 8) and for each ∈ {2, 3}, the spectrum of all the integers such that there is a handcuffed design of order and block size in a 4-cycle system of order is determined in [5]. In this paper we want to complete the case = 4 by determining the set of all the integers such that there is a path design of order and block size 3 in a 4-cycle system of order