Almost resolvable minimum coverings of complete graphs with 4-cycles

If the complete graph Kn has vertex set X, a minimum covering of K with 4-cycles, (X,C, P), is a partition of the edges of K ∪ P into a collection C of 4-cycles, where P is a subgraph of λK and the number of edges in P is as small as possible. An almost parallel class of a minimum covering of K with 4-cycles is a largest possible collection of vertex disjoint 4-cycles (so with ⌊n/4⌋ 4-cycles in it). In this paper, for all orders n, except order 9 which does not exist, we exhibit a minimum covering of K with 4-cycles so that the 4-cycles in the covering are resolvable into almost parallel classes, with any remaining 4-cycles being vertex disjoint. We also complete the missing examples of order 23 for the same problem with almost resolvable maximum packings with 4-cycles, and orders 41 and 57 in the case of an exact decomposition into 4-cycles, partitioned into almost parallel classes