5-cycle systems of [formula omitted] and [formula omitted].

In Bryant et al. (1996) it is shown that there exists a 5-cycle system of K v + u − K v if and only if the obvious necessary conditions are satisfied. One purpose of this paper is to extend this result by providing the necessary and sufficient conditions for the existence of a 5-cycle system of λ K v + u − λ K v . It was shown in Asplund et al. (2013) that if there exists a 5-cycle system of λ K v , then there exists a 5-cycle system of ( λ + m ) K v + 1 − λ K v 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 λ K v . In doing so, we will show that there exists a 5-cycle system of ( λ + m ) K v + 1 − λ K v if and only if the obvious necessary conditions are satisfied, except possibly in two cases. [ABSTRACT FROM AUTHOR]