Almost resolvable maximum packings of complete graphs with 5-cycles

Abstract

Let X be the vertex set of K n . A k-cycle packing of K n is a triple (X, C, L), where C is a collection of edge disjoint k-cycles of K n and L is the collection of edges of K n not belonging to any of the k-cycles in C. 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 K n , denoted by k-RMCP(n), is a resolvable k-cycle packing of K n , (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 = 3, 4 has been decided. When n ≡ k (mod 2k) and k ≡ 1 (mod 2) or n ≡ 1 (mod 2k) and k ∈ {6, 8, 10, 14} ∪ {m: 5 ⩽ m ⩽ 49, m ≡ 1 (mod 2)}, D(n, k) also has been decided with few possible exceptions. In this paper, we shall decide D(n, 5) for all values of n ⩾ 5.