On the existence of holey 4-cycle frames

A (k,λ)(k,λ)-HCF(n,mu)(n,mu) is a pair (C,R)(C,R) where CC is a kk-cycle decomposition of the λλ-fold of the Cayley graph Γ(G,S)Γ(G,S) with G=Zu×Zm×ZnG=Zu×Zm×Zn and (x,y,z)∈S(x,y,z)∈S if and only if x≠0≠zx≠0≠z, and RR is a partition of CC into holey parallel classes each of which consists of mn(u−1)k cycles whose vertices cover all GG except one suitable coset of {0}×Zm×Zn{0}×Zm×Zn. In this paper, we shall give some recursive constructions for (k,λ)(k,λ)-HCFs, and show that there exists a (4,λ)(4,λ)-HCF(n,mu)(n,mu) if and only if mn(u−1)≡0(mod4), λ(n−1)m≡0(mod2), n≥2n≥2, u≥3u≥3 and (λ,n,u,m)≠(2r,2,3,2t+1)(λ,n,u,m)≠(2r,2,3,2t+1), r≥1,t≥0r≥1,t≥0.