Almost Resolvable Maximum Packing Of Complete Graphs With5-Cycles

Let X be the vertex set of Kn. A k-cycle packing of Kn is a triple (X, C, L), where C is a collection of edge disjoint k-cycles of Kn and L is the collection of edges of Kn not belonging to any of the k-cycles in C. An almost parallel class of a k-cycle packing of Kn is a collection of [n/k] vertex disjoint l-cycles. When n≡0(mod k), an almost parallel class is said to be a parallel class. A k-cycle packing (X, C, L) is called resolvable if C can be partitioned into almost parallel classes. A resolvable maximum k-cycle packing of Kn, denoted by l-RMCP(n), is a resolvable k-cycle packing of Kn (X, C, L) in which the number of almost parallel classes is as large as possible.Let D(n, k) denote the number of almost parallel classes in a k-RMCP(n). D(n, k) for k=4has been decided by Billington et al. recently. When n≡k (mod2k) and k≡1(mod2)or n≡1(mod2k) and k∈{6,8,10,14}∪{m:5≤m≤49, m≡1(mod2)}, D(n, k) also has been decided by Niu et al. with few possible exceptions. In this paper, we shall decide D(n,5) for all values of n≥5.