5-cycle systems of (λ+m)Kv+1−λKv and λKv+u−λKv

It was shown in Asplund et al. (2013) that if there exists a 5-cycle system of λKv, then there exists a 5-cycle system of (λ+m)Kv+1−λKv if and only if v is (λ,5)-admissible. The second purpose of this paper is to extend this result by removing the requirement that there exists a 5-cycle system of λKv. In doing so, we will show that there exists a 5-cycle system of (λ+m)Kv+1−λKv if and only if the obvious necessary conditions are satisfied, except possibly in two cases.